Bh

II

J < { 4 '£ \ 1 ^-\V' |Au' , '/U) ( C ! * if 'i i^ r'

A^

>**/»*;

X^S

v>^'

i v

7 < ^

v ^csL'

Kikrvrs*

^.OTfsrSpsr^*^

' r /iy

&&

& 1 'mm

' N * -

m^^^i

i"/WP

^ >k .kK. i-Jtfflt

*" - _ yv^

^0

r^ rv :i

2 :%^ >■

> ?

;*/^y*y

SpW^##!

"' ► * * .

A /V.

/'

,A-

'tfe^w

*ws*-

!^*M«I^

i*VM!!3£®ff

"$$f?

2^T*»l

rat*

/«#^^a&|^^%

AO/T^'

Ait ^

5^*2*^23

*4*>V

vuv

fcfcltk

g.n.^

TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

ESTABLISHED November 15, 1819.

VOLUME THE SEVENTH.

CAMBRIDGE:

PRINTED AT THE UNIVERSITY PRESS;

AMD BOLD BY

JOHN WILLIAM PARKER, WEST STRAND, LONDON;

J. & J. J. DEIGHTON ; AND T. STEVENSON, CAMBRIDGE.

MDCCC.XLII.

CONTENTS OF THE SEVENTH VOLUME.

Part I.

PAUK

N°. I. On the Laws of the Reflexion and Refraction of Light at the common Surface

of two non-crystallized Media: by George Green, B.A. of Caius College .... 1

II. On Molecular Equilibrium. Part I. By the Rev. Philip Kelland, M.A., Queens'

College, Cambridge ; Professor of Mathematics in the University of Edinburgh. 25

III. On Rolling Curves: by Hamnett Holditch, M.A., Fellow of Cakes College. 6l

IV. Note on the Motion of Waves in Canals: by Geohge Green, B.A. of Caius

College 87

V. On the Nature of the Molecular Forces which Regulate the Constitution of the

Luminiferous Ether: by S. Eabnshaw, M.A. of St John's College 97

VI. Supplement to a Memoir on the Reflexion and Refraction of Light: by G. Green,

B.A. of Caius College 115

Part II.

VII. On the Propagation of Light in Crystallized Media: by G. Green, B.A.

Fellow of Caius College 121

VIII. On a Portion of the Tertiary Formations of Switzerland : by D. T. Ansted,
M.A., Fellow of Jesus College. Fellow of the Society and, of the Geological
Society; Professor of Geology in King's College, London 141

IX. On the Quantity of Light intercepted by a Grating placed before a Lens; and
on the Effect produced by Interference: by the Rev. Philip Kelland, M.A.,
F.R.SS.L. & E., late Fellow of Queens' College, Cambridge ; Professor of
Mathematics in the University of Edinburgh 153

X. On the Foundation of Algebra, No. I.: by Augustus De Morgan, F.R.A.S.,
F.C.P.S., of Trinity College; Professor of Mathematics in University College,
London 173

XI. On the Effect of the Non-Residence of Landlords, fyc. on the Wealth of a

Community : by J. Toze r, M. A., of Caius College 1 89

XII. Demonstration that all Matter is heavy: by the Rev. William Whewell,

B.D., Fellow of Trinity College and Professor of Moral Philosophy 197

XIII. On the Position of the Axes of Optical Elasticity in Crystals belonging to the

Oblique-Prismatic System: by W. H. Miller, M.A., F.R.S., Fellow and
Tutor of St John's College, and Professor of Mineralogy in the University
of Cambridge ' 209

XIV. On a New Construction of the Going-Fusee : by G. B. Airy, Esq., Astronomer

Royal 217

IV

CONTENTS.

N". XV.
XVI.

XVII.

XVIII.
XIX.
XX.

XXI.

XXII.
XXIII.

XXIV.

Part III.

PAGE

On Spurious Rainbows: by W. H. Miller, M.A., F.R.S., Professor of
Mineralogy in the University of Cambridge 277

On the Foundation of Algebra, No. II.: by Augustus De Morgan, F.R.A.S.,
F.C.P.S., of Trinity College ; Professor of Mathematics in University Col-
lege, London 287

An Enquiry into the Causes which led to the Fatal Accident on the Brighton
Railway (Oct. 2, 1841), in which is developed A Principle of Motion of
the greatest importance in guarding against the Disastrous Effects of Col-
lision under whatever circumstances it may occur : by the Rev. J. Power,
M.A., Fellow and Tutor of Trinity Hall, Cambridge 301

Discussion of the Question : — Are Cause and Effect successive or simultaneous ?
by the Rev. William Whewell, B.D., Master of Trinity College, and
Professor of Moral Philosophy 319

On the Motion of a small Sphere acted upon by the Vibrations of an Elastic
Medium : by the Rev. James Challis, M.A., Plumian Professor of Astro-
nomy in the University of Cambridge 333

Description of an Extinct Lacertian Reptile, Rhynchosaurus artieeps, Owen,
of which the Bones and Foot-prints characterize the Upper New Red
Sandstone at Grinsill, near Shrewsbury: by Richard Owen, F.R.S., G.S.
fyc, Hunterian Professor in the Royal College of Surgeons 355

A general Investigation of the Differential Equations applicable to the Motion
qf Fluids : by the Rev. James Challis, M.A., Plumian Professor of
Astronomy and Experimental Philosophy in the University qf Cambridge .. . 371

On the Propagation qf Luminous Waves in the Interior qf Transparent Bodies :
by the Rev. M. O'Brien, M.A., late Fellow of Caius College, Cambridge 397

On the Steady Motion qf Incompressible Fluids : by G. G. Stokes, B.A.,
Fellow of Pembroke College 439

On the Truth qf the Hydrodynamical Theorem, that if udx + vdy + wdz
be a Complete Differential with respect to x, y, z, at any one instant, it
is always so: by the Rev. J. Power, M.A., Fellow and Tutor qf Trinity
Hall 455

ADVERTISEMENT.

The Society as a body is not to be considered responsible for any
facts and opinions advanced in the several Papers, which must rest
entirely on the credit qf their respective Authors.

The Society takes this opportunity of expressing its grateful
acknowledgements to the Syndics of the University Press, for their
liberality in taking upon themselves a portion of the expense of printing
this Part of its Transactions.

TRANSACTI ONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

Volume VII. Part I.

CAMBRIDGE:

PRINTED AT THE PITT PRESS;

AND SOLD BY

JOHN WILLIAM PARKER, WEST STRAND, LONDON;

J. & J. J. DEIGHTON ; AND T. STEVENSON, CAMBRIDGE.

M.DCCC.XXXIX.

I. On the Laws of the Reflexion and Refraction of Light at the common
Surface of two non-crystallhed Media. By George Green, Esq.,
B.A., Caius College.

[Read December 11, 1837-]

M. Cauchy seems to have been the first who saw fully the
utility of applying to the Theory of Light those formulae which re-
present the motions of a system of molecules acting on each other by
mutually attractive and repulsive forces ; supposing always that in the
mutual action of any two particles, the particles may be regarded as
points animated by forces directed along the right line which joins
them. This last supposition, if applied to those compound particles, at
least, which are separable by mechanical division, seems rather restrict-
ive ; as many phenomena, those of crystallization for instance, seem
to indicate certain polarities in these particles. If, however, this were
not the case, we are so perfectly ignorant of the mode of action of
the elements of the luminiferous ether on each other, that it would
seem a safer method to take some general physical principle as the
basis of our reasoning, rather than assume certain modes of action,
which, after all, may be widely different from the mechanism employed
by nature; more especially if this principle include in itself, as a par-
ticular case, those before used by M. Cauchy and others, and also
lead to a much more simple process of calculation. The principle
selected as the basis of the reasoning contained in the following paper
is this: In whatever way the elements of any material system may
act upon each other, if all the internal forces exerted be multiplied
by the elements of their respective directions, the total sum for any
Vol. VII. Part I. A .

2 Mr GREEN, ON THE REFLEXION

assigned portion of the mass will always be the exact differential of
some function. But, this function being known, we can immediately
apply the general method given in the Mecanique Analytique, and
which appears to be more especially applicable to problems that re-
late to the motions of systems composed of an immense number of
particles mutually acting upon each other. One of the advantages of
this method, of great importance, is, that we are necessarily led by
the mere process of the calculation, and with little care on our part,
to all the equations and conditions which are requisite and sufficient
for the complete solution of any problem to which it may be applied.

The present communication is confined almost entirely to the con-
sideration of non-crystallized media; for which it is proved, that the
function due to the molecular actions, in its most general form, con-
tains only two arbitrary coefficients, A and B ; the values of which
depend of course on the unknown internal constitution of the medium
under consideration, and it would be easy to shew, for the most gene-
ral case, that any arbitrary disturbance, excited in a very small portion
of the medium, would in general, give rise to two spherical waves,
one propagated entirely by normal, the other entirely by transverse,
vibrations, and such that if the velocity of transmission of the former
wave be represented by y/A, that of the latter would be represented
by y/B. But in the transmission of light through a prism, though the
wave which is propagated by normal vibrations were incapable itself of
affecting the eye, yet it would be capable of giving rise to an ordinary
wave of light propagated by transverse vibrations, except in the ex-

A A

treme cases where ■== = 0, or — = a very large quantity ; which, for

the sake of simplicity, may be regarded as infinite; and it is not diffi-
cult to prove, that the equilibrium of our medium would be unstable

A 4

unless j5 > - . We are therefore compelled to adopt the latter value of

-=:, and thus to admit that in the luminiferous ether, the velocity of

transmission of waves propagated by normal vibrations, is very great
compared with that of ordinary light.

AND REFRACTION OF LIGHT. 3

The principal results obtained in this paper, relate to the intensity
of the waves reflected at the common surface of two media, both for
light polarized in and perpendicular to the plane of incidence; and
likewise to the change of phase which takes place when the reflexion
becomes total. In the former case, our values agree precisely with
those given by Fresnel; supposing, as he has done, that the direction
of the actual motion of the particles of the luminiferous ether, is
perpendicular to the plane of polarization. But it results from our
formulae, when the light is polarized perpendicular to the plane of
incidence, that the expressions given by Fresnel are only very near
approximations; and that the intensity of the reflected wave will
never become absolutely null, but only attain a minimum value; which,
in the case of reflexion from water at the proper angle, is ^ P art
of that of the incident wave. This minimum value increases rapidly,
as the index of refraction increases, and thus the quantity of light re-
flected at the polarizing angle, becomes considerable for highly refract-
ing substances, a fact which has been long known to experimental
philosophers.

It may be proper to observe, that M. Cauchy {Bulletin des Sciences^
1830,) has given a method of determining the intensity of the waves
reflected at the common surface of two media. He has since stated,
{Nouveaux Exercices des Mathematiques,) that the hypothesis employed
on that occasion is inadmissible, and has promised in a future memoir,
to give a neiv mechanical principle applicable to this and other questions ;
but I have not been able to learn whether such a memoir has yet ap-
peared. The first method consisted in satisfying a part, and only a part,
of the conditions belonging to the surface of junction, and the con-
sideration of the waves propagated by normal vibrations was wholly over-
looked, though it is easy to perceive, that in general waves of this kind
must necessarily be produced when the incident wave is polarized perpen-
dicular to the plane of incidence, in consequence of the incident and
refracted waves being in different planes. Indeed, without introducing
the consideration of these last waves, it is impossible to satisfy the Whole
of the conditions due to the surface of junction of the two media.
But when this consideration is introduced, the whole of the conditions

A2

■4 Me GREEN, ON THE REFLEXION

may be satisfied, and the principles given in the Mecanique Analytique
became abundantly sufficient for the solution of the problem.

In conclusion, it may be observed, that the radius of the sphere of
sensible action of the molecular forces has been regarded as unsensible
with respect to the length X of a wave of light, and thus, for the
sake of simplicity, certain terms have been disregarded on which the
different refrangibility of differently coloured rays might be supposed to
depend. These terms, which are necessary to be considered when we
are treating of the dispersion, serve only to render our formulas uselessly
complex in other investigations respecting the phenomena of light.

Let us conceive a mass composed of an immense number of molecules
acting on each other by any kind of molecular forces, but which are
sensible only at insensible distances, and let moreover the whole system
be quite free from all extraneous action of every kind. Then x y and %
being the co-ordinates of any particle of the medium under consideration
when in equilibrium, and

x + u, y + v, z + iv,

the co-ordinates of the same particle in a state of motion (where u, v,
and w are very small functions of the original co-ordinates (x, y, %) of
any particle and of the time (/)), we get, by combining D'Alembert's
principle with that of virtual velocities,

5; ' D »•{S'*'' + £ J '' + ^H =SZ,, '• ^ * (1);

Dm and Dv being exceedingly small corresponding elements of the mass
and volume of the medium, but which nevertheless contain a very great
number of molecules, and $<j> the exact differential of some function
and entirely due to the internal actions of the particles of the medium
on each other. Indeed, if ^0 were not an exact differential, a perpetual
motion would be possible, and we have every reason to think, that the
forces in nature are so disposed as to render this a natural impossibility.

AND REFRACTION OF LIGHT. 5

Let us now take any element of the medium, rectangular in a state
of repose, and of which the sides are dx, dy, dz the length of the
sides composed of the same particles will in a state of motion become

dx' — dx (1 + Si), dy'— dy(\ + s 2 ), dz' '■= d*(l + # 3 ) 5

where s t , s 2 , s 3 are exceedingly small quantities of the first order. If,
moreover, we make

dy' „ dx' dx

a = cos<^,, fi-vm< Mt 7 = cos< rfy ;

a, fi and 7 will be very small quantities of the same order. But, what-
ever may be the nature of the internal actions, if we represent by

cS(p dx dy dx,

the part of the second member of the equation (1), due to the molecules
in the element under consideration, it is evident, that <p will remain the
same when all the sides and all the angles of the parallelopiped, whose
sides are dx dy' dz', remain unaltered, and therefore its most general
value must be of the form

(p = function \t u s if s 3 , a, fi, y}.

But 8 U s 2 , s 3 , a, /3, 7 being very small quantities of the first order,
we may expand cp in a very convergent series of the form

(p = (p + </>, + (p 2 + 03 + &c. :

<po, <pi, &> &c. being homogeneous functions of the six quantities
a, fi, 7, g u s if s 3 of the degrees 0, 1, 2, &c. each of which is very great
compared with the next following one. If now, p represent the
primitive density of the element dx dy dz, we may write p dx dy dz
in the place of Dm in the formula (1), which will thus become, since
<p is constant,

fffp dx dy dn^Su + ^U + tLg j w J

= fffdx dy dz (50, + $(p 2 + &c.) ;

the triple integrals extending over the whole volume of the medium
under consideration.

b Mn GREEN, ON THE REFLEXION

But by the supposition, when u = 0, v = and w = 0, the system is
in equilibrium, and hence

= fffdx dy da $fa :

seeing that 0, is a homogeneous function of s lf #*, s 3 , a, /3, 7 of the
, first degree only. If therefore we neglect fa, fa, &c. which are ex-
ceedingly small compared with <p 2 , our equation becomes

fffp dx dy dz i^ iu + ( -~ It, + -^ SwJ = fffdx dy dz Sfa (2) ;

the integrals extending over the whole volume under consideration.
The formula just found is true for any number of media comprised in
this volume, provided the whole system be perfectly free from all
extraneous forces, and subject only to its own molecular actions.

If now we can obtain the value of fa, Ave shall only have to apply
the general methods given in the Mecaniquc Analytique. But fa, being
a homogeneous function of six quantities of the second degree, will in
its most general form contain 21 arbitrary coefficients. The proper
value to be assigned to each, will of course depend on the internal
constitution of the medium. If, however, the medium be a non-crystal-
lized one, the form of fa will remain the same, whatever be the
directions of the co-ordinate axes in space. Applying this last con-
sideration, we shall find that the most general form of fa for non-
crystallized bodies contains only two arbitrary coefficients. In fact, by
neglecting quantities of the higher orders, it is easy to perceive that

_ du _ dv dw

ax dy dz

dw dv _ div du du dv

dy dz' dx dz' T'T" dy dx'

and if the medium is symmetrical with regard to the plane (xy) only,
fa will remain unchanged when — z and — to are written for z and w.
But this alteration evidently changes a and j8 to — a and — /3. Similar
observations apply to the planes (xz) (yz), If therefore the medium is
merely symmetrical with respect to each of the three co-ordinate planes,
we see that fa must remain unaltered when

AND REFRACTION OF LIGHT.

or — «, — w, —a, — /3
or - y, - v, -a, - 7
or — x, — u, — (3, — 7

8, w, a, /3
are written for \ y, v, a, y
x, u, /3, 7,

In this way the 21 coefficients are reduced to 9, and the resulting
function is of the form

*&

+ H (%

l(~y + La !! + M^ + Ny 2

or ,dv dw Q du dw „ du dv

dy ' dx dx' dz dx ' dy ^ 2 '

(A).

Probably the function just obtained may belong to those crystals
which have three axes of elasticity at right angles to each other.

Suppose now we further restrict the generality of our function by
making it symmetrical all round one axis, as that of 8 for instance.
By shifting the axis of x through the infinitely small angle $6 ;

and

X

x + ydU

y

i becomes

y-xSe,

8

8

d '

dx dy

dx

d
dy

; becomes ,

— -de—

dy dx '

d

d

dz j

. dz

u

u + vSO

V

• becomes '

v-uSe .

w .

IV

Making these substitutions in (A), we see that the form of 2 will
not remain the same for the new axes, unless

L = M,
P=Q;

(B).

8 Mb GREEN, ON THE REFLEXION

and thus we get

under which form it may possibly be applied to uniaxal crystals.

Lastly, if we suppose the function 2 symmetrical with respect to all
three axes, there results

L = M= N,
P=Q =#;

and consequently,

^ [dy ' d% dx' dz dx' dy\' >

or, by merely changing the two constants and restoring the values of
a, /3, and 7,

, (du dv dw\-
^ 2 \dx dy d»J

(C).
7 i/du dv\~ (du dw\* (dv dwy .(dv dw du dw du dv\\
\\dy dx] Was dx) \dx dy) \dy' dz dx' dz dx'dy))'

This is the most general form that (p 2 can take for non-crystallized
bodies, in which it is perfectly indifferent in what directions the
rectangular axes are placed. The same result might be obtained from
the most general value of 2 , by the method before used to make <p 3
symmetrical all round the axes of z, applied also to the other two axes.
It was, indeed, thus I first obtained it. The method given in the text,
however, and which is very similar to one used by M. Cauchy, is not
only more simple, but has the advantage of furnishing two intermediate
results, which may possibly be of use on some future occasion.

AND REFRACTION OF LIGHT. 9

Let us now consider the particular case of two indefinitely extended
media, the surface of junction when in equilibrium being a plane of
infinite extent, horizontal (suppose), and which we shall take as that
of (yz), and conceive the axis of x positive directed downwards. Then
if p be the constant density of the upper, and p t that of the lower
medium, 2 and 2 " the corresponding functions due to the molecular
actions. The equation (2) adapted to the present case will become

fffp dx dy d% tgp Su + -~ lt> + -^ lw\

+ fffp. d * d y d % \jf * «, + rfjr *•, + -$■ 3 ™ j , (3).

= fff dx d V d% & + fffdx dy dz <$> ;

u t , v t , w t belonging to the lower fluid, and the triple integrals being
extended over the whole volume of the fluids to which they respectively
belong.

It now only remains to substitute for 2 and ffi their values, to effect
the integrations by parts, and to equate separately to zero the coefficients
of the independent variations. Substituting therefore for <p 2 its value (C),
we get

fffdx dy dx S(p 2
. rrr , , » f (du dv dw\ [d$u d$v d$w\ )

= - A ^ dxd y d% \{Tx + Ty + ^)\^ + ~^ + -d^))

J? /7T/7 // // \(dw dv\ (dlu d$v\ (du dw\ ld§u d$w\
•'•'•' * \\dy dx) \dy dx) \d% dx) \d% dx I

(dv dw\ (dSv d§w\ r/rf» d$w dw dly \

\d% dy) \d% dy) L \dy d% d%' dy )

(du d§w dw d%u\ (du d$v dv dSu \ -i 1
\dx' d% d%' dx] \dx dy dy' dx/jj

Vol. VII. Paet I. B

10 Mr GREEN, ON THE REFLEXION

/ttj j 7 i 4 d (du dv dw\ n rd 2 u d 2 u d fdv dw\-\] .

/ a d (du dv dw\ jfTd^v d 2 v d /du div\~]\ .
\ «ty ' Wa? dy dz J L rfir das 2 dy \dx dz) -1 f

\ a d_ [du rfv dw \ rd 2 w d 2 w d (du dv\~\ 1 .

1 dz'Kdx dy dz) L dx 2 dy 2 dz' \dx dy) J J "*'

seeing that we may neglect the double integrals at the limits x = — ee ,
/y = +co, s = + oo; as the conditions imposed at these limits cannot
affect the motion of the system at any finite distance from the origin ;
and thus the double integrals belong only to the surface of junction, of
which the equation, in a state of equilibrium, is

* x.

In like manner we get

fffdxdydzW?

+ the triple integral;

since it is the least value of x which belongs to the surface of junction
in the lower medium, and therefore the double integrals belonging to
the limiting surface, must have their signs changed.

If, now, we substitute the preceding expression in (3), equate sepa-
rately to zero the coefficients of the independent variation 8u, §v, Sw,
under the triple sign of integration, there results for the upper medium

AND REFRACTION OF LIGHT. 11

d*u . d_ idu dv dw\ _ (d 2 u d"u d idv dw\ 1
p ~df ~ A dx'\dx + dy + dz) + [dy 1 + ~d% 2 ~ dx' Xjdy^ ~d%)y

d 2 v . d idu dv dw\ -n\d 2 v d 2 v d idu dw\)
9 ~dt 2 ~ dl/'\dx' + dy + ~dj) + \dx 1 + dz T ~d^'(d^ + 'd^)f' (4);

d 2 w _ . d idu dv dw \ „ ( d 2 w d 2 w d fdu dv\i
p dt 2 ~ dz'\dx dy d%) \dx 2 dy 2 dz'\dx dy)]''

and by equating the coefficients of Su /} Sv t , $w t , we get three similar
equations for the lower medium.

To the six general equations just obtained, we must add the con-
ditions due to the surface of junction of the two media; and at this
surface we have first,

u = u t , v = v t , w = w t , (when x = 0), (5);

and consequently,

%u = $u t ; 8v = Sv t ; §w = Sw,.

But the part of the equation (3) belonging to this surface, and which
yet remains to be satisfied, is

♦/r**{*.(£*&)***(&*&W«

and as 8u = Su t , &c, we obtain, as before,

. idu dv dw\ „ idv dw\ . ldu t dv^ dw \ __ „ idv^ dw \

\dx dy dz J \dy dz J ' \dx dy dz) ' \dy d% I

„ idu dw\ _ ldu / dw \ _

\d% dx) ~ ' \d% dx J '

and these belong to the particular value x = 0.

B2

12 Mr GREEN, ON THE REFLEXION

The six particular conditions (5) and (6), belonging to the surface
of junction of the two media, combined with the six general equations
before obtained, are necessary and sufficient for the complete determi-
nation of the motion of the two media, supposing the initial state of
each given. We shall not here attempt their general solution, but
merely consider the propagation of a plane wave of infinite extent,
accompanied by its reflected and refracted waves, as in the preceding
paper on Sound.

Let the direction of the axis of %, which yet remains arbitrary, be
taken parallel to the intersection of the plane of the incident wave
with the surface of junction, and suppose the disturbance of the particles
to be wholly in the direction of the axis of as, which is the case with
light polarized in the plane of incidence, according to Fresnel. Then
we have

= u, = v, = m , = v/,
and supposing the disturbance the same for every point of the same
front of a wave, w and tv l will be independent of %, and thus the three
general equations (4), will all be satisfied, if

d 2 w „ (d 2 w d 2 w)

p lf =B Xd^ + -df}'

ft

or by making — = 7 s ,

d 2 w . (d 2 w d 2 w)

dt ■ '' \S* ^Wl' (7) "

Similarly in the lower medium we have
d 2 w, 2 ld 2 w, d 2 w}

f 'Xd¥ + dy 2 }' (8) '

df
w t and 7, belonging to this medium.

It now remains to satisfy the conditions (5) and (6). But these are
all satisfied by the preceding values provided,

w = w n

rydw _ „ dw,

dx ' dx '

AND REFRACTION OF LIGHT. 13

The formulae which we have obtained are quite general, and will
apply to the ordinary elastic fluids by making B — 0. But for all the
known gases, A is independent of the nature of the gas, and conse-
quently A — A r If, therefore, we suppose B = B lt at least when we
consider those phenomena only which depend merely on different states
of the same medium, as is the case with light, our conditions become*

w = w t \
dw^ _ dw\ (when x = 0), (9).
dx dx)

The disturbance in the upper medium which contains the incident
and reflected wave, will be represented, as in the case of Sound, by

w —f{ax + by + ct) + F( — ax + by + ct);

f belonging to the incident, F to the reflected plane wave, and c being
a negative quantity. Also in the lower medium,

«, -/(«,* + by + ct).

These values evidently satisfy the general equation (7) and (8), pro-
vided c* = 7 2 (a* + ¥), and c 2 = y* {aj + 5*) ; we have therefore only to
satisfy the conditions (9), which give

f(by + ct) + F (by + ct) =/(Jy + cf),

af (by + ct) - aF' (by + ct) - «,/' (by + ct).

Taking now the differential coefficient of the first equation, and
writing to abridge the characteristics of the functions only, we get

v-(i+?)jr. -*w-(i-J)/;

* Though for all known gases A is independent of the nature of the gas, perhaps it is
extending the analogy rather too far, to assume that in the luminiferous ether the con-
stants A and B must always be independent of the state of the ether, as found in different
refracting substances. However, since this hypothesis greatly simplifies the equations due to
the surface of junction of the two media, and is itself the most simple that could be selected,
it seemed natural, first to deduce the consequences which follow from it before trying a more
complicated one, and, as far as I have yet found, these consequences are in accordance with
observed facts.

14 Mr GREEN, ON THE REFLEXION

and therefore

1- ^

F[ a _ a - a, _ cot 9 - cot 9, _ sin {9- 9) #

J" a t ~ a + a t ~ cot 9 + cot 9 t ~ sin (9 / + 9) '

a

9 and 9 t being the angles of incidence and refraction.

This ratio between the intensity of the incident and reflected waves,
is exactly the same as that for light polarized in the plane of incidence,
(vide Airy's Tracts, p. 356,) and which Fresnel supposes to be propagated
by vibrations perpendicular to the plane of incidence, agreeably to what
has been assumed in the foregoing process.

We will now limit the generality of the functions f, F and f t , by
supposing the law of the motion to be similar to that of a cycloidal
pendulum; and if we farther suppose the angle of incidence to be in-
creased until the refracted wave ceases to be transmitted in the regular
way, as in our former paper on Sound, the proper integral of the
equation

d i w l , j d 2 w t d 2 w}

IF ~ 7/ 1 das' + If) '
will be

w, = e- a '' x Bsm^, (10);
where ^ = by + ct, and a' is determined by

7 /(6 2 -«; 2 ) = ^ = y(* 2 +« 2 ), (ii).

But one of the conditions (9) will introduce sines and the other
cosines, in such a way that it will be impossible to satisfy them unless
we introduce both sines and cosines into the value of w, or, which
amounts to the same, unless we make

w = a sin {ax + by + ct + e) + /3 sin ( - ax + by + ct + e), (12),

in the first medium, instead of

w — a sin (ax + by + ct) + (5 sin ( — ax + by + ct),
which would have been done had the refracted wave been transmitted
in the usual way, and consequently no exponential been introduced into

AND REFRACTION OF LIGHT. 15

the value of w r We thus see the analytical reason for what is called
the change of phase which takes place when the reflexion of light
becomes total.

Substituting now (10) and (12), in the equations (9), and proceeding
precisely as for sound, we get

= a cos e — /3 cos e t ,

= a sin e + fi sin e t ,

— B = a sin e — 8 sin e ,
a

B = a cos e + /3 cos e t .

Hence there results a = 0, and e t = — e, and

tan e = — = -j- •£ t = -f tan 9.
a b b b

But by (11),

5
b

if-^-^-^Fii-y^-OT)*

by introducing ^ the index of refraction, and 6 the angle of incidence.
Thus,

to n,- *^' rin ''- 1 > ;

M COS

and as e represents half the alteration of phase in passing from the
incident to the reflected wave, we see that here also our result agrees
precisely with Fresnel's, for light polarized in the plane of incidence.
(Vide Airy's Tracts, p. 362.)

Let us now conceive the direction of the transverse vibrations in
the incident wave to be perpendicular to the direction in the case
just considered ; and therefore that the actual motions of the particles
are all parallel to the intersection of the plane of incidence (xy) with
the front of the wave. Then, as the planes of the incident and re-
fracted waves do not coincide, it is easy to perceive that at the
surface of junction there will, in this case, be a resolved part of the
disturbance in the direction of the normal; and therefore, besides the

16 Ma GREEN, ON THE REFLEXION

incident wave, there will, in general, be an accompanying reflected and
refracted wave, in which the vibrations are transverse, and another pair
of accompanying reflected and refracted waves, in which the directions
of the vibrations are normal to the fronts of the waves. In fact, unless
the consideration of the two latter waves is also introduced, it is im-
possible to satisfy all the conditions at the surface of junction ; and these
are as essential to the complete solution of the problem, as the general
equations of motion.

The direction of the disturbance being in plane (xy) w = 0, and as
the disturbance of every particle in the same front of a wave is the
same, u and v are independent of * Hence, the general equations (4)
for the first medium become

d 2 u _ 2 d idu dv\ , d idu dv\
df ~ dx \dx dy) ' dy \dy dx) '

d*® _ 2 d (du_ dv\ 2 d fdv du\
~d? =S d~y \dx + dy) +y dx[dx~dy)'

where g* = — , and y 2 = — •

These equations might be immediately employed in their present
form; but they will take a rather more simple form, by making

d<p dty
dx dy '

(13).
d(p d\f/ 1
dy dx '

<p and ^ being two functions of x, y and t, to be determined.

By substitution, we readily see that the two preceding equations are
equivalent to the system,

d? ~* Ux 2 + dy 2 )'

(14).

df " 7 \dx* + dy* I '

AND REFRACTION OF LIGHT. 17

In like manner, if in the second medium we make

' dx dy '

(15).

v - d( f > ' d ^>
' dy dx '

we get to determine <p, and >|/ / the equations

df *' \da* dy 2
^ _ * * ( d l±i x £*<

(16).

df " \dx* dtf

and as we suppose the constants A and B the same for both media,
we , have

7, g,'

For the complete determination of the motion in question, it will be
necessary to satisfy all the conditions due to the surface of junction of
the two media. But, since w = and w t = 0, also, since u, v, u t , v, are
independent of as, the equations (5) and (6) become

u = u t , v = »,;

\dx dy) dy \dx dy I dy

du dv _ du, dv,
dy dx dy dx '

provided x = 0. But since x = in the last equations, we may differ-
entiate them with regard to any of the independent variables except x,
and thus the two latter, in consequence of the two former, will become

du _ du, dv _ dv,

dx dx' dx dx'

Substituting now for u, v, &c, their values (13) and (15), in <p and \f/,
the four resulting conditions relative to the surface of junction of the
two media may be written,

Vol. VII. Part I. C

18

Mr GREEN, ON THE REFLEXION

dx dy

dj, = d<f>, d^ y
dx dy

d(p
dy

d^f _ d(p /
dx dy

dx

d 2 d 2 ^ m £& j gjfc
d# 2 da; dTy

d 2

V (when a; = 0) ;

rfo; 2

dxdy

d> _ d>, d>,
da; 2 d# dy

da; dy da; 2 dx dy dx 2

or since we may differentiate with respect to y, the first and fourth
equations give

d 2 ^ , d 2 ^_d 2 f dS//,

+

+

dx 2 dy 2 dx 2 T dy 2 '
in like manner, the second and third give

d 2 <p d 2 (p
dx 2 + dy 2

d 2 0, + d'fr

dx 2 ' dy 2 '
which, in consequence of the general equations (14) and (16), become

^NL-._^ flm l ** - d% *>
y 2 df ■ 7/ 2 df g 2 dt 2 ~ gfdf

Hence, the equivalent of the four conditions relative to the surface
of junction, may be written

d<p d^/ _ d(f> t d^ l
dx dy dx dy

d(p d^> _ d(p t dy\r t
dy dx dy dx

d 2 <p d 2 (p,
g*df- gfdf

d> d>,

7,

> (when x = 0),

7

df

2 df

(17).

If we examine the expressions (13) and (15), we shall see that the
disturbances due to <p and <f> / are normal to the front of the wave to
which they belong, whilst those which are due to ^ and ^ are trans-
verse or wholly in the front of the wave. If the coefficients A and B

AND REFRACTION OF LIGHT. 19

did not differ greatly in magnitude, waves propagated by both kinds of
vibrations must in general exist, as was before observed. In this case,
we should have in the upper medium

\// = f {ax + by + ct) + F{— ax + by + ct),
and (18).

<P — X ( ~ a 'x + by + ct) ;

and for the lower one

^,=f, («> + h + ct),

(19).

The coefficients b and c being the same for all the functions to
simplify the results, since the indeterminate coefficients a' a t a' will
allow the fronts of the waves to which they respectively belong, to
take any position that the nature of the problem may require. The
coefficient of x in F belonging to that reflected wave, which, like the
incident one, is propagated by transverse vibrations would have been
determined exactly like a] a t d, as, however, it evidently = — a, it was
for the sake of simplicity introduced immediately into our formulae.

By substituting the values just given in the general equations (14)
and (16), there results

(f = ( a * + b 2 ) 7 2 = (a* + ¥) 7/ 2 = (a' 8 + ¥)g 2 = («/ 2 + b*)g%

we have thus the position of the fronts of the reflected and refracted
waves.

It now remains to satisfy the conditions due to the surface of
junction of the two media. Substituting, therefore, the values (18) and
(19) in the equations (17), we get

" § " .

x = -p x, ;

-a' x ' + b(f' + F') = a; x ; + bf;,
b x ' - a(f - F') = b x ; - aj; ■;

where to abridge, the characteristics only of the functions are written.

C2

20 Mr GREEN, ON THE REFLEXION

By means of the last four equations, we shall readily get the values
of F"'x" f"x" m terms of f", and thus obtain the intensities of the
two reflected and two refracted waves, when the coefficients A and B
do not differ greatly in magnitude, and the angle which the incident
wave makes with the plane surface of junction is contained within
certain limits. But in the introductory remarks, it was shewn that

A

■= = a very great quantity, which may be regarded as infinite, and

therefore g and g t may be regarded as infinite compared with 7 and y r
Hence, for all angles of incidence except such as are infinitely small,
the waves dependent on <p and <p t cease to be transmitted in the regular
way. We shall therefore, as before, restrain the generality of our
functions, by supposing the law of the motion to be similar to that
of a cycloidal pendulum, and as two of the waves cease to be transmitted
in the regular way, we must suppose in the upper medium

\J, = a sin {ax + by + ct + e) + /3 sin ( — ax + by + ct + e),
and (20).

(p = e ax (A sin \f/ + B cos \J/ ) ;

and in the lower one

v// = a sin (a x + by + ct),

(21).
<t>, = e a,x {A, sin ^ + B t cos ^ ),

where to abridge \// = by + ct.

These substituted in the general equations (14) and (15), give
c« = 7 2 {a* + P) = 7; {a? + b>) =g*(- a' 2 + b 2 ) =g?{- «/ 2 + b 2 ),
or, since g and g, are both infinite, N

b = a' = a;.

It only remains to substitute the values (20) (21) in the equations
(17), which belong to the surface of junction, and thus we get

bA sin \//o + bB cos ^ + ba cos (>// + e) + bfi cos (\|/ + e)

m — bA t sin \f/ — bB t cos \// + ba / cos >|/- »

AND REFRACTION OF LIGHT. 21

bA cos v//o - bB sin \J/- - aa cos (>//■<, + e) + «/3 cos (\|/ + 0,)

= bA t cos \^ — i B t sin v// — a,^ cos >^ , (22).

— {A sin \|/- + B cos \|> ) = — (A, sin \js + -B, cos ^ ),

1 1

— \a sin (v|/ + e) + /3 sin (^ + e)\ = — i a l sin >/,„.
7 7/

Expanding the two last equations, comparing separately the coefficients
of cos >// and sin ^ , and observing that

ff y

— = — = m suppose,

o/ 7/

we get

# = m 2 ^,

(23).
o cos e + /3 cos e, = /**<*,»

a sin e + /3 sin e, = 0.

In like manner the two first equations of (22) will give

= A + A / — a sin e — /3 sin e t ,

= A — A, + -j- 1 + t (p cos e t — a cos e),

= 2? + -B, + a COS + /3 COS 0, — a /5

= -B — -B, + t (/3 sin <?, — a sin e) ;
combining these with the system (23), there results

= A + A t ,

= B + B, + (fx 2 - l),a,,
aa, a

(24).

= A - A, + -jr-' + t (/3 COS 9, - a COS e),

= 2? — B t + v (/3 sin e t - a sin e).

22 Mb GREEN, ON THE REFLEXION

Again, the systems (23) and (24) readily give

a Sin e = — A . -"Hj -^ a.

and therefore

/3 s

a COS e = ^ . f m 2 + — j a ,
_ . . (m 2 - l) 2 6

fieose = |. (m 2 - J) «,;

,.+ ir ,(„.-sy + () ,.- 1) .g

"^v + i)..(„. + |)' + („'-ir^

(25).

(26).

When the refractive power in passing from the upper to the lower
medium is not very great, p. does not differ much from 1. Hence, sin e
and sin e t are small, and cos e, cos e t do not differ sensibly from unity ;
we have, therefore, as a first approximation,

a sin 2 9 cot 9,

/3 " a

sin 2 Q t cot 9

sin 29 — sin 29,
~ sin 20 + sin 29,

_ tan (0 - 9)

a

sin 2 9 cot 9-

— +

sin 2 9 cot 9

™ tan (0 + ) '

which agrees with the formula in Airy's Tracts, p. 358, for light polar-
ized perpendicular to the plane of reflexion. This result is only a near

approximation : but the formula (26) gives the correct value of Ej , or

the ratio of the intensity of the reflected to the incident light; sup-
posing, with all optical writers, that the intensity of light is properly
measured by the square of the actual velocity of the molecules of the
luminiferous ether.

From the rigorous value (26), we see that the intensity of the re-
flected light never becomes absolutely null, but attains a minimum
value nearly when

AND REFRACTION OF LIGHT.. 23

= /u 2 , i. e., when tan (9 + 9) — to ,

Co

which agrees with experiment, and this minimum value is, since (27)

b
gives - - M ,

£-, (w } " 2 0'-> v (28) .

4(/u + 1) fx* + (m - 1) -|

4
If /u. = - , as when the two media are air and water, we get
3

- = — nearly.

It is evident from the formula (28), that the magnitude of this
minimum value increases very rapidly as the index of refraction in-
creases, so that for highly refracting substances, the intensity of the
light reflected at the polarizing angle becomes very sensible, agreeably
to what has been long since observed by experimental philosophers.
Moreover, an inspection of the equations (25) will shew, that when we
gradually increase the angle of incidence so as to pass through the po-
larizing angle, the change which takes place in the reflected wave is not
due to an alteration of the sign of the coefficient (5, but to a change
of phase in the wave, which for ordinary refracting substances is very
nearly equal to 180°; the minimum value of /3 being so small as to cause
the reflected wave sensibly to disappear. But in strongly refracting sub-
stances like diamond, the coefficient /3 remains so large that the re-
flected wave does not seem to vanish, and the change of phase is con-
siderably less than 180°. These results of our theory appear to agree
with the observations of Professor Airy. (Comb. Phil. Trans. Vol. iv.
p. 418., &c.)

Lastly, if the velocity 7, of transmission of a wave in the lower
exceed 7 that in the upper medium, we may, by sufficiently augment-
ing the angle of incidence, cause the refracted wave to disappear, and
the change of phase thus produced in the reflected wave may readily
be found. As the calculation is extremely easy after what precedes, it

24 Mr GREEN, ON THE REFLEXION, &c. OF LIGHT.

seems sufficient to give the result. Let therefore, here, n = — , also

7
e, e t and 6 as before, then e t — — e, and the accurate value of e is
given by

*. /~n — T75 Tn (m 2 — l) 2 tan 6

tan e - n Vfi* tan 2 9 - sec 2 - K — — - z - .

M + 1

The first term of this expression agrees with the formula of page 362
Airy's Tracts, and the second will be scarcely sensible except for highly
refracting substances.

II. On Molecular Equilibrium. Part I. By the Rev. Philip Kelland,
M.A., Queens'" College, Cambridge; Professor of Mathematics in
the University of Edinburgh.

[Read March 26, 1838.]

INTRODUCTION.

1. Whatever ideas may have been entertained of the nature of
forces at a distance from the centre of action, there appear to have
been no very definite notions current respecting molecular forces, till
within a few years from the present time. The obvious change in
the attractions of the different parts of a solid body, produced by sepa-
rating the particles by ever so small an interval ; the fact that the
attraction of cohesion when destroyed cannot be restored by any
ordinary pressure, indicated that the force which the particles exert on
each other in their positions of equilibrium, is of a nature totally distinct
from the appreciable attractions and repulsions at finite distances. New-
ton only threw out hints respecting the nature of forces of this kind,
never applying them, except in a popular manner in his Optics. One
kind of molecular force which he conjectures is that of the, particles
of air and the magnetic ones, Newton applies to calculation, but he
by no means supposes his hypothesis the correct one; on the contrary
he appears to entertain great doubts on the subject, for he concludes
his scholium by observing: "Whether elastic fluids do really consist
of particles so repelling each other, is a physical question. We have
demonstrated the properties of fluids consisting of particles of this kind,
that hence philosophers may take occasion to discuss that question."
Vol. VII. Part I. D

26 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

The phenomena of electricity and magnetism did indeed suggest
hypotheses respecting the internal constitution of bodies, but these
hypotheses, for the most part, were only partial ones. Those of
iEpinus, Cavendish, and Franklin, fully establish a disposition of dif-
ferent sets of particles, but leave the possibility of such a disposition
as consistent with the conditions of equilibrium to other hypotheses of
a nature totally different from the one applied. With one or two ex-
ceptions it would appear, that all writers have regarded the molecular
force as of a nature either distinct from that of the attractions and
repulsions of the electric particles, or as the fundamental expression
of which the law, in the latter case, is only a limiting form.

About the middle of the last century, however, Dr Knight published
his " Attempt to explain all the Phenomena of Nature by means of
two Principles, Attraction and Repulsion." The hypothesis adopted in
this work, appears to be nearly the same as that usually adopted by
theorists in Chemistry of the present day, and which is not essentially
different from that which forms the basis of the present Memoir, with
the exception, that the Author supposes the law of force to be the
inverse power of the distance. Bodies are imagined to be formed
of combinations of two groups of particles acting differently on each
other, the one set mutually attractive, the other mutually repulsive;
the former, by peculiar arrangements aggregated together, determine
the nature of different substances ; whilst the latter are collected around
these groups, and form their atmospheres. I regret that I have not
been able to meet with Dr Knight's work, which appears from the
notices of it, to have been a sound and admirable treatise.

A few years later appeared Boscovich's " Theoria Philosophic Natu-
ralis ad unicam legem virium, in Naturd existentium redacta" a work
which from its title professes the reduction of all forces to one and
the same law. As that law will be found to be the conclusion from
another more simple law, I shall briefly state its principal features.

(l). "The atoms of matter are endued with attractive or repulsive
forces to one another, of which the law of variation is the same for
all."

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 27

(2). "Action and reaction are equal."

(3). "The nature of the force is such that at different distances
it is attractive and repulsive alternately, so that a particle in reced-
ing from another, is first repelled, then attracted, then again repelled,
and so on."

(4). " When the distance is indefinitely diminished, the force is re-
pulsive and is indefinitely increased; and when the distance is inde-
finitely increased, the force is attractive and diminishes as the inverse
square of the distance."

Such are the general features of Boscovich's law of molecular action.
It will be our endeavour to deduce from an hypothesis not very dif-
ferent from that of Knight, a law resembling the above in its general
features.

2. Notwithstanding the long interval that has elapsed since the
publication of Boscovich's work, very little has been done on the sub-
ject, except by way of application, until very lately. Capillary attrac-
tion is a phenomenon, the solution of which, clearly requires a molecular
hypothesis ; but, unfortunately, the nature of the question is such that
it is satisfied without the aid of any specific restriction to the law,
except that it should be one which very rapidly diminishes as the dis-
tance increases, and is insensible at distances appreciable by our senses.
Hence, we know that Laplace in his Mecanique Celeste, and Poisson
after him, have not cared to assume any particular law of force, and
even if they had, no means would have been found for its verification.
One result of this fact appears to be, that the circumstance of an
active force of this nature being sufficient to explain a phenomenon
totally different in character from those of cohesion and combination, by
which it is obviously suggested, induced Laplace himself to the belief
that this was the ultimate law. If such be not the case, I am unable
to account for his adoption of such a law, absurd as it appears, in
his explanation of the phenomena of heat. It would have been sup-
posed, that this was an opportunity of applying the beautiful analysis
of the former parts of his work to the reduction of the molecular law
to some simple form. But such is not the case, nor does Poisson, even

D 2

28 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

in the Memoir where he detects the insufficiency of Laplace's hypo-
thesis of capillary attraction, attempt to ascend higher in the investiga-
tion. In his Theory of Heat too, he introduces discrete molecules only
for the purpose of generalizing the problems of conduction and ra-
diation, without attempting to solve those of expansion and crystal-
lization : so that he makes no progress whatever in the explanation of
phenomena.

3. In my Memoir on Dispersion, I endeavoured to shew that the law
of the inverse square of the distance, is that of the attraction or repulsion
of the particles of light, and in subsequent Memoirs, I have endeavoured
to reduce some of the phenomena of sound and heat to the same
law. Nothing, however, was effected with respect to the equilibrium
of the molecules. The latter object has lately been accomplished, at
least partially, by M. Mossotti, in a Memoir "On the Forces which
regulate the Internal Constitution of Bodies." The hypothesis of Mossotti
is the same as Dr Knight's, except that the forces vary inversely as
the square of the distance. It is proved, that one set of particles may
have an atmosphere of another set, the density of which varies rapidly
in receding from the surface of the former. M. Mossotti then endeavours
to find the conditions of equilibrium of a particle of the first or the
material set. It is ion this point that I conceive M. Mossotti's hypothesis
completely fails. The law of action of two particles as deduced by
M. Mossotti, is composed of two parts, a repulsive part which vanishes
when the distance is sensible, and an attractive part which varies in-
versely as the square of the distance. Now when it is borne in mind
that the whole set of forces acting on any particle must be sufficient
to retain that particle in equilibrium at a certain distance from the one
next to it, we shall perceive that this law of action requires that the
mutual distance, or the density of the particles, should vary as the
magnitude of the body. I do not mean to assert, that the density
should be increased in the same ratio as the mass is increased, but that
it must be so increased, that the repulsive force of the adjacent particles
should be very nearly in the proportion of the linear magnitude of the
body. I cannot think this a probable, hardly a possible, condition of
matter. The state of the surface may depend, and probably does so.

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 29

on the thickness of the solid, provided the solid be very thin, but it
can scarcely be conceived to do so in other cases, much less to vary
equally with the superficial extent of the surface itself.

4. In order then to determine as nearly as possible, what is the
law of distribution of the particles of caloric, or the universally diffused
system of particles, as well as what is the law of aggregation of the
material particles, which determines whether the arrangement have the
properties of elasticity, fluidity, solidity, crystalline arrangements, &c.
I have examined a number of different arrangements, and investigated
the conditions of their equilibrium and stability.

In the present part, I have said little about the application to dif-
ferent states of consistence, deeming it more prudent to make a series
of calculations in the first place. In fact, it is most probable that the
forms of the results will in all cases, as they certainly are in those
I have already tried, be very different from those which a popular view
of the subject would suggest. In my treatise on Heat, however, will
be found some applications roughly stated, which I hope more fully to
investigate in the sequel.

Investigation of the Conditions of Equilibrium.

5. I purpose to commence my investigation, by retaining M. Mos-
sotti's hypothesis of two systems of particles repulsive towards atoms
of their own kind, but each respectively attractive towards the atoms
of the other. We will call one system of particles caloric, and the
other matter; the masses of the atoms of the former being very small
compared with those of the latter. We will suppose the former dis-
tributed through space, whilst the latter occupy only certain given
positions: in both, the density at different points will be essentially dif-
ferent, but the particles of the latter medium, will in all cases be sup-
posed wherever they exist, to be much more widely separated than those
of the former, so that a material particle may be considered as a nucleus,
about which the particles of caloric are collected, so as to form its
atmosphere.

30 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

To find the conditions of equilibrium of a particle of caloric.

6. Let x, y, z be the co-ordinates of a particle of caloric measured
from any point as origin, x, y, % those of another particle, D the
density of the caloric estimated by the number of particles in a given
volume in the neighbourhood of the former particle : D' the correspond-
ing quantity for the latter; let also X, Y, Z be the co-ordinates of
a particle of matter supposed spherical, and collected at its centre of
gravity in all cases in which its own attraction or repulsion is to be
calculated; call P the mass of an atom of caloric, M that of an atom
of matter, each estimated by the attraction or repulsion exerted by it
on a unit of either caloric or matter at the distance unity : let V be
the sum of each particle of caloric, divided by its distance from that
whose co-ordinates are x, y, z; U the sum of each particle of matter
divided by its distance from the same point ; also, let r be the former
distance, R the latter corresponding to the particles respectively, whose
co-ordinates are x ', y, %'; X, Y, Z, then

y = p rrr dx ' dy'dz-H

R

I have adopted integrals for the caloric, as it is supposed that the
particles are so near each other, that the variation of action due to the
.situation of a particle with respect to those immediately surrounding it,
forms no important element in the calculation. I shall have occasion
to mention this subject more explicitly in the sequel.

In order to fix the ideas, let it be supposed that x', y, z; X, Y, Z
are in advance of xy%, so that

r = \/V - xf + {y' — yf + (%' - zf,

R = V(X- xf + (Y- yf + (Z- %f ;

then the action of the caloric on the particle in question parallel

dV
to the axis of x is P. -7—, and since the force is repulsive, it tends to

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 31

diminish x; for a like reason, that of the matter on the same particle
is jP.-t— tending to increase x; consequently the whole force with which

the particle is urged in the direction of the axis of x is P i—j j— ].

7. By the substitution of integrals in the place of sums, the ex-
pression V, as before noticed, is no longer the total action of the caloric
on the particle, subject as it is to the powerful variations of action of
these particles by which it is immediately surrounded ; it is in fact,
the total action, omitting these and corresponding variations for the
other particles. In order to obtain the conditions of equilibrium of
the particle, we must apply another force, viz. the variation of action
due to the place of the particle.

Without entering into calculation respecting this force, it is evident
at once, that its value is increased in the same ratio as the increment

of the density at that point, and must consequently vary as —*— ; but

whether it might not also vary as D, does not appear so obvious. The
following investigation is perhaps more satisfactory.

8. Conceive a portion of the mass to form a prism* of which the
axis is parallel to x. Let its section be unity, and its length Sx, and
suppose the caloric within it to have the uniform density D, then the
action on it, due to the above forces, is

pm*^-^):

\dx ax )

let p be the pressure on the end next the origin, p + -^- Ix + &c. that
on the further end, then we must have

dp - PD (— - — ) •

dx \dx dx J '

here, then, by taking the aggregate of a large nnmber of particles,
we eliminate the effect of the molecular variations which retain any
individual one in its place, and may consider p as the actual pressure
exerted, by whatever means it matters not, to retain the particles

32

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

which form one end of the prism in their places. Now the surround-
ing particles will produce this effect, and it is obvious that the
action on any individual particle will vary as the number of particles
which act on it, supposing the positions left out of consideration. Thus,
suppose («) particles occupying certain positions to exert a force F, then
if two particles could be supposed to occupy the place of each one,
the force would become %F, and so on. Under these circumstances,
then, the repulsion on an individual particle would vary as the density,
and whatever be the mode of arrangement, the same law appears the
most simple and probable. Similar reasoning applies to the density of
the particles acted on, and we conclude that p oc Lf,

Let p = \ elf ;

dp r.dD

dx dx '

t , dD P (dU dV\

and then -j— = — -j j— ,

dx c \dx dx J

d*D P idU d 2 V

dx'

~ ~c \d& ~ ~d¥ 1 '

d 2 D P id*U
dtf c \ dtf

d*D P [d*U

dz 2

-f( !

dz"

d 2 V \
dtf)

drr

dz 2

(i);

but

d 2 V d"V d*V . __

-d¥ + dy* + ~d*=~^ Pn

d 2 U d 2 U d 2 U

+ S-T + —r^r =

dx 2

dtf

dz 2

(2).
(3).

. d'D d 2 D d 2 B

hence -j^- + -x-* +

dx 2

dtf

dz 2

= -f(

P id 2 V d 2 V d'V

dx 1

dtf dz

r)

4ttP 2

if we designate

c

4ttP 2

D

(4).
by « 2

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 33

10. The solution of this equation will be found in various Memoirs
of M. Poisson and others: it is

4g-aV(i,-i)»+(j r ,)« + (j,-0»

Z> = 2

the symbol 2 having reference to points whose co-ordinates are x t , y t , % r

With respect to the points in question there can be no difficulty,
for, from the form of the solution, it is evident that the medium is
influenced symmetrically with respect to any such points, and moreover,
the solution of (2) will give V a function of the same quantities,
whence equation (1) will determine U to be a function of the same;

M

but the value of U being 2 -=* it is obvious that all the quantities

are functions only of R, or of \/{X— x) 2 + (Y—yf + (Z — %f,
hence the value of D becomes

^ e -a\/(X-x)* + (r-y)*+(Z-z)>

Z> = 2

= 2

Vrx-xf + (r- y y + <z- %y

Ae- aR
B '

11. It may be remarked, that in this solution, as in the corres-
ponding one in my treatise on Heat, I have departed slightly from
M. Mossotti's Memoir, by considering the attraction or repulsion of
a particle on another, to be proportional to the product of their masses,
so that P 2 , MP, M 2 are respectively the forces exerted at the distance
unity, by P acting on P, P acting on M, and M acting on M. The
reasoning of M. Laplace, to which M. Mossotti refers for the proof
of the theorem that the pressure varies as the square of the density,
I have not retained, as I conceive it does not take notice of the real
point of difficulty ; namely, that the force which constitutes the right-
hand side of the equation, is not the force on any individual particle,
unless the sums are so expressed as to indicate the small variations due
to the rapid change of action of those particles immediately surround-
ing that acted on. Indeed, were they the real actions, their sum would
Vol. VII. Part I. E

34 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

doubtless be zero. This difficulty will be surmounted by taking U and
V, as I have done, not for the absolute forces, but for the sum of all the
forces if the particles were distributed symmetrically: the pressure, in
this case, which arises from the action of the particles contiguous to that
under consideration, will obviously vary as the square of the density.

12. I proceed in the next place, to determine the value of V, by
two different methods, the comparison of which, will prove that no
inaccuracy arises from the adoption of direct variation in deducing the
equation (2), provided such variation is known to be uninterrupted.

To integrate equation (2) we must observe that

ft 2 ?.— r/ 2 S— rf 2 ?.—

a2j B a *R R _ n

~inr + dy* + d%*

and must consequently include 2-^ in the value of V in the place and
with the interpretation of an arbitrary constant.

_ . d*V d 2 V d*V \ ^Ae-"*

We have -d^ + -dy + ^ = ~ ^^-R--

The complete value of V will therefore be

C being given by the equation

Co 2 = 4ttP;

Ae~ aR \ 4ttP

tr-v\ B Ae-° R \ 47T.

13. In order to calculate the value of V directly, it is most con-
venient to employ polar co-ordinates; the particle M being the pole.
Let A be the place of the particle acted on, P that of any other
particle, AP = r, AM = R, angle PMA = 9, MP = P ; radius of a
particle = /, then 2 7rp 2 sin 0d6dp is the volume of an elementary
annulus ;

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 35

JJ pr

rr 2irp sin 9 e- a P dQ dp p

M Vp 2 + R*-2pBcosd

m s Ap j ^e-^p^^-2Bpcose + ^ df>

For the portion included within a spherical surface, whose centre is
M and radius MA, we must have the limits 8 = 0, 8 = w and p less
than R\ for the remainder p is greater than R;

hence ^ = ZAP.f R ?£ dpe""> {R + P -(R- P )\

+ %AP. r^dpe-"r{p + R-(p-R)}

m ^ Ap rn ^pe-^dp I ^ Ap r- 4wr ., d

- -9 a p \ ^ e ~ al (1 + <**) _ 4tt e— R \

-^ Ar \R~- a 2 a 2 ^RJ

The coefficient -» being the same as obtained by the other method,

a

shews that equation (2) is correct; we also perceive that

B = Ae~ al (l +al).

14. The equation (1) will give a relation between a, c and the
other quantities which we proceed to investigate.

From the value of D (10),

^ = 2^ («-* + «**-•*) (X-*),

also ~ = ^2^(B-Ae- aR -AaRe-' R )(X-x),

E 2

36 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

hence the equation gives

2 4 (e- aR + «Re~ aR ) {X - x)

B 3

\M 4nrP

~ 7 ■ 2 S " Sf-»-'««- , ' + -*«"'>} (>-*).

whence it evidently follows that

KB? " ^^j ~ '

and — =1.

Co 2

The last equation merely verifies the operation, since the value of a
which it gives, is no other than its assumed value in (Art. 9).

The other equation gives M = —

cB

~ P

PM

but from the nature of c, it evidently varies as P\ call it therefore
aP 2 , where a is a quantity independent both of M and P; the result
is

, 4tt
a = ^'

B ~^P'

or a is the same for all substances, whilst B varies as the attractive
energy of the particle of matter.

15. This conclusion is of great importance, as it enables us to cal-
culate the effect of any individual particle independently of those by
which it is accompanied. In fact, whatever be the nature of the mass,
any individual particle will be surrounded by an atmosphere of caloric,

g-aR

the density of which varies as — p— , where B is the distance from its

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 37

centre; whilst the density at a given distance varies only as the attractive
energy of the particle. Of course, the expression density does not signify,
the actual amount of aggregation of particles, but merely the aggre-
gation so far as it depends on the particle under consideration.

16. We may verify our conclusion with respect to the value of e,
by the following method :

Conceive only one particle to exist. At a considerable distance R
from its centre, the principal forces which act on a particle of its
surrounding caloric, are the attraction of the particle and the repulsion
of the caloric.

MP

The former force is -g-,- .

™ . . 4ntP*J r ar , 4nrP*A . „ r 1

The latter ■— «; — .)e~ ar rdt = — ^ — \C - - e~ ar - -e~ ar \,

R* ' J R 4

a a

the value of which from r = I to r = a considerable quantity is very
nearly

4,*P*R MP

hence

Ra* ~ W '
. birPB

" " M '
the same value as we obtained by the former process.

17. We have hitherto omitted any consideration of a uniform layer
of caloric distributed over space, so as to act equally on every point.
It is clear, that the effect of such caloric will be found by retaining D
as the excess of density above this uniform density q. The correct
value of V will now be found by subtracting from its value above
the sum of every mass displaced by a material molecule divided by
its distance from the point under consideration ; hence, all we have to
do is to diminish V by a quantity

- 2,q ' 3- R~ 3 * P **'B'

38 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM,

and B = Ae- at (l+al)-^-;

1+al 3 1+al'

The equations in (14) are not affected by this consideration, conse-
quently B is independent of the mean density ; and A is increased
proportionally to it.

18. I propose next to determine the mutual action of two particles.

We have seen that the atmosphere of any particle is perfectly in-
dependent of that of the surrounding particles : it follows, that the action
of two particles on each other, is also independent of the surrounding
medium. The latter supposes, however, that the pressure which is
exerted by the caloric is due to the actions of particles so arranged
as to produce equilibrium ; in fact, the pressure on the surface of a
material particle A, even as far only as it depends on the caloric which
constitutes the atmosphere of B, will vary with the attractions of the
other particles on it, except the system be in equilibrium, in which
case we may suppose, as we have already done, that the pressure cor-
responding to the density Doc D

= hD.

19. Our first point will be to find the value of this pressure.

Let a be the distance between the centres of the particles, / their
radius, P any point in the particle on which the pressure is to be de-
termined; then the area of an annulus is QirPsmedQ,

p—aR

and the pressure on it ZwPhA —p- sinfleW;

hence, the resolved part of the whole pressure in the direction of the
line joining the centres of the particles, is

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 39

2irl 2 hA / . — — .dd = Q,

* \/a* + P- 2al cosd

h sin 0<7fl r/jg

vV + P - 2al cos " al '

„ ■• 2irlhA r e- aR dR{a 2 + l*- R 2 )
- 1*^ /(«* + U - R 2 ) e-« R dR,

the limits being R = a — I, and R = a + I;

The attraction of the caloric is (16 and 17) very nearly,
4,ttMPA fl ni I al 1 aa a \ 4tt o/WP

« la 2 a a 2 a J 3 a''

whilst the mutual repulsion of the two particles is —j-;

hence, the expression for the whole force of mutual attraction of the
particles towards each other, is

4>ttMPA [e~ al 1,1 a ] M 2 4>tt qPMP

&m — — s ) +- e -«i -~e-" a --e~ aa ) - — - — ^ — -, —

a \ a 2 a a a ) OT 3 a 2

- *-¥ ft.^4) «-t - (- 7 + ^4)--i-

20. Here we have not taken into consideration the circumstance
that the mass of the particle will not be acted on exactly as if collected
at its centre of gravity. It has been supposed that it is so collected,
and that the caloric then extends to infinity, so that the attraction is
due to a quantity of caloric lying in a sphere about the attracting
particle at the distance of the attracted one. Now, in fact, nearly one
half the attracted particle will not be acted on so much by the laminse
beyond its surface, whilst the other portion is actually acted on by
particles beyond the laminae at the centre; but as the density of the

40 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

former laminae is greater, and the part of the body on which it acts
less, we cannot have erred much in taking the mean as the correct
value of the attraction.

Indeed, if there be any error committed, it is obvious that we have
estimated the attraction too high ; both from the greater density being
that which we have supposed to have full agency, and from the fact,
that the actual attraction on parts lying at a distance from the centre,
is not in the direction of the line joining the centres of the particles.
It may then be conceived, that the above expression is rather too great
for the attraction, and it will appear presently that its value even as
I have given it, is negative.

iirPB

For we have already proved (14) that , — = M;

.-. — , — (\+al)e- al —J? = M (17);

a o

hence

s = — 7^r~ (1 + aa) e " ~^~ e t ° * (1 + aa)t

an essentially negative result.

21. We may however introduce a positive quantity into this ex-
pression, by conceiving each molecule as a compound one of two
different kinds of particles attracting each other, as we proceed to
shew.

M'
Let U'=^~,

then the action on a particle of caloric is

, (dU dU' dV

P -T— +

dx dx dx
hence all the equations for motion are unaffected :

and Z> = 2 — „ — + 2 — ~ — ,

|S Ae~« R 1? A'e- aR '\ 4ttP
V ~ * \R ~ B + B " ' R ) « 8 ;

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 41

. M „ dD P (dU dU dV\
hence the equation, , = — I —j— + -7 -j— 1

ClX C \ ClX (I X ClX /

gives 2^ (e~' R + aRe~ aR ) + 2 ^ («-««' + aRe~* R ')

c 2, lR 3 + B' 3 a* \R> + R> AKe +aUe

- A'(e-° R ' + aRe-° R )\~\ ,

hence M + M' = — — ^-= .

2

a

The expression for the attraction of a particle of the first substance
on one of the same kind, whose mutual distance is a, is

„ _ IttMPA f (l +al)e- al (1 +aa)e- aa \ M*

a 8 I a 2 a' ~)~ IF

and a similar expression, only accenting the letters, is true for the
mutual attractions of the other similar particles; call it S' : also, since
M attracts M', if T be the attraction of M' in virtue of M, we shall
have

rr _^rPM'A i<J+al)e-"' (l+aa)e- aa \ MM'

1 ^ \ J " a* ) + ~aT

_ 2*hA e _ aa | ^Vyal + 1 j g _ ae _ &c J f
T - *" PMA ' f (l+qf)g— r (l+aa)g-'"' l 3fJT

V \ a* a* J +_ ^~

2ttA^' „ ifa + l + aal V
e —

a*
Vol. VII. Pa»t I.

~{F^*a-'-<4

42 PROFESSOR KELLANB, ON MOLECULAR EQUILIBRIUM.

hence, the whole mutual attraction of a compound particle is

S + ST + T + T

= ^J*{BM+B'M'-(l + aa)e-<"'{AM + A'M')}- W-MJ

-^e-« a l\A + A'\{e-" l (a + l+aal + ±) - e*' l .fa - I - aal + i)
+ e- al ' (a + l')+ }].

Now for a single particle M = — , and there is no reason to

suppose B different in other cases ; hence, the mutual attraction of the
compound particles is

i^' - ^(AM + A'M')(1 + aa)e-°°

-2mtf e ' aa{A + A '"> ^' al ' {a + l)+ - *

Now if they are at a distance from each other, the quantity e~ aa is

2 MM'
very small, and the force is ; — varying inversely as the square of

the distance, which is the known law of gravitation.

22. I shall not dwell longer on this point, as the difficulty is not
to obtain a portion of the expression which shall vary inversely as the
square of the distance; for this will be at once accomplished either by
the above method, or by supposing the attraction of M on P a little
greater than MP, as M. Mossotti has done, or by taking into the
calculation the caloric which is displaced by a particle, either by the one
attracting, or that acted on, which in accuracy ought to be done. But
the difficulty is to obtain an expression for the mutual action of two
particles, which shall express those facts of Boscovich's hypothesis
specified in the Introduction, and which are clearly essential to the
nature of a molecular action.

To accomplish this object, I have supposed all the particles repul-
sive; which hypothesis requires that the density of the caloric within

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 43

the medium, be less than that without. I shall not attempt to justify
this hypothesis, or to prove that its apparent complexity, as compared
with the received one, affords a strong a priori argument against its
correctness. The only way to obtain final accuracy, is to subject to
rigid calculation any hypothesis which may suggest itself, and to retain
that which gives results consistent with facts. And should it be found
that a little difficulty attaches itself to the one in question, we may ex-
pect either that the difficulty itself will vanish, or the hypothesis will be
found unnecessary from after-attention to a more simple one. I may
state, that I have spent a considerable portion of time in trying other
hypotheses, but at present can find none which so apparently coincides
with known phenomena as that which I have just stated.

23. Let us then determine the conditions of equilibrium of a
system in which the atoms of caloric are repulsive to those of matter.

Assume the density of the external caloric to be q, and that of the
internal q', so that by writing q — q for D, we may adapt some of our
previous investigations to this case.

where V has reference to every particle.

But p = \cq*\

dp , da'

' = co — —

dx * dx *

d£ = P fdU dV\
dx c \dx dx I '

dD _ dq^ _ d£ ' P fdU + dV\ + ety
dx ~ dx dx c \ dx dx) dx'

Now if there were no material particle, we should have

dx c dx '

where V t is the quantity which V becomes for a homogeneous medium
of density q\ if then we assume V I -V=V; where V is the function

F2

44 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

due to a mass of particles equal to those displaced by the repulsion,
and situated in the places from which they have been driven off; we
shall get

dD P (dU _ dV\

dx c \ dx dx

dx 2 c \ dx 2 dx'

"-)■•

P id 2 U d 2 V'\

V

f l (TV' &'V d'V'\
dx^ '*"dy T "*" W e \~dx r + ~df + ~dtf~) ;

d 2 D d 2 D d 2 D P (d 2 V d 2 V d 2 V

and — r- 5- + — r? +

, v d 2 V d 2 V d 2 V , _ ,

but -da? + df + dz 2 =-^ P <l>

d 2 V, d 2 V, d 2 V t : _

d 2 (V-V) d 2 {V-V) d*{V-V) A h . k

•'■ dx 2 + dy 2 + M ; =-^(?-g)>

d 2 V d 2 V d 2 V' D _

or iM + iitf + -dir = -^ PD >

d 2 D d 2 D dTD 4ttP 2
• • rf* 2 + rfy 2 + d% 2 ~ c '"'

= a 2 D;
the solution of which equation is

Ae« R .
R '

and, as in the former case, it evidently follows that

B Ae~ aR \ 4ttP

\R R J " a 2

24. By employing a process precisely analogous to that in (13),
we obtain the value of V directly, taking into the account the caloric
displaced by the material particles ; the expression is

' ~ a 2 ^1 R R ) + 3 ' ?n i'

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 45

hence B = Ae~ al {\ + al) + ^— -£,

3

and we obtain also, as in (14),

%/r cB 4,irPB

,, ,. " , a 2 i)if a 2 / 3 ?

4,rP r 3 )'

4>^AMPe- al (l+al) = M*_^ 4>7rMl 3 Pq
a 2 « 8 « 3 3 a 2 '

which expressions will simplify that for the force of two material par-
ticles on each other, by striking out several identical terms.

7b find the mutual action of two particles of matter together with
the caloric surrounding them, on the hypothesis that matter is repulsive
towards caloric.

25. Since the caloric surrounding the particle A, whose action on
B we are about to estimate, is diminished by ^4's repulsion, the ex-
ternal mass will no longer produce an effect equal in all directions,
whose actual value is therefore zero ; but will exert a force on B
equivalent to the attraction of a mass similar, and similarly situated to
the mass displaced.

The set of forces, then, which act on B through the means of A, are

(i). The repulsion of A on B.

(2). The attraction of a mass of caloric equal to that displaced by
the volume of A.

(3). The attraction of a mass of caloric equal to that displaced by
the repulsion of A, and

(4). The pressure on the surface of B resolved in one direction
along the line joining the centres of A, B.

26. If a be the distance between the centres of A and B, the

M 2
expression for their repulsion is — 5- , which is the first force.

46 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

4nrl 3

27. The value of the second force is also 2 .qMP, calling the
exterior or mean density q.

28. To obtain the third force, we must divide the displaced caloric,
or rather a portion equal to it placed in a position directly opposite,
into two portions; the one containing all that is included in a sphere
whose centre is the centre of A, and radius the distance to that point
of B which is nearest to A ; the other, the portion arising from the
spherical shell included between two surfaces to radii equal to the
distances of the nearest and most distant point of B, from the centre
of A.

The former of these is easily found as in (16), equal to

4* MP A , (1 + ^ _ e _ a(a _ t) I3| + a {a _ l ^

a 2 a L ' "

To obtain the latter, we will first omit the consideration of the
portion which would occupy the place of B, supposing that particle
removed, and consequently take no notice of the quantity which ought
to be displaced there; by this means, it is obvious that we shall estimate
the attraction a little too highly; and we shall see that the portion,
taken as we have supposed, is actually less than would be obtained by
conceiving the mass of B collected at its centre ; consequently the whole
attraction is considerably less than that given in (20). Now we saw
that the resultant action even on that calculation was essentially nega-
tive, it appears then that a more rigorous analysis increases rather than
diminishes the difficulty attendant on an attractive atmosphere of caloric.

29. Let us then proceed to the calculation.

The action of a mass of caloric in a spherical shell of thickness 8B,
whose radius is R on a similar portion of a shell of the body B at
radius p, is easily seen to be*

AMP 4nrR*$Be- aR , rfa . , . .. ''?<
v . ^ • UJ 2 f sin d P d( t> cos 0) ;

* For the construction of the figure, &c. see the Note (a) at the end.

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 47

of which the last factor

= /(-^cos20 + C)

= fa dp (1 - cos 2<p) = irdp sin 2 <p

= "\P- J i^y d P}

+ 2(a 2 - P)(a + I - R)

+ <* + %-* }]:

consequently the whole attraction is

AM ^ V " fdRe- aR \(a + l-R)R--^ («T7 a~^T . a + l-R)

-^ (a*-P)(a + l-R)R- {a+ ^: :R3 R}
2a i v ' v ; 12a

-\

\a 2 )

a* + l* m R>
R +

2« s " ' 12<

AMP 4,**e-« R ((a*-ly 2 . 7W . . 72X (\+aR\

« +/ : /i? 2 2R 2\ 1 /jR 4 41F 12R* 24 J? 24\1

48 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

AMP i^e—

V

*r* [ e - a i i^jzll _ A. (a * + P) !+«(« + /)

a 2 + l 2 ( 12 2a + 1 2\ 1 f——j* ia + l 12a + l

24 a + l 24\ 1
. . tta*-l*y 2 ,. 73 . /1+ao^A

a* + l*( /2 2a-/ 2\ 1 / - j4

2 V o a 2 / 12 V

a* + Pf H 2a-/ 2\ 1 /• , 4 4a-/ 12a-/

— - +-

24 a - I 24

The quantity under the bracket following e - "', becomes by addition

a*-2a 2 l 2 + l i a 2 + l 2 , 2 2a 2 a 1 + aa , 72V

-. + s (a 2 + — + - ; + 2 . / + I 2 )

4 2 a a 2 a '

1 ( ( 4 a 3 12 a 2 24 « 2 a
12 a a a a

/. , 12a 2 24a , 24\ 7 /_ , 12a 12\ „

+ (ta + ^.P + l 4 } - -V (a 3 + / 3 ) (1 + aa + a/)

\ a) 5 3 a a v v

a 4 a 2 ( , 2a 2\ 1 / , 4a 3 . 12a 2 24a 24\
4 2 \ a a 2 / 12 V a a 2 a a 4 /

/l + aa 11+ ao\ „. (1 1 1 \ 74 2 , , 7 , N ,_ 7 ,

\ a 3 o / \4 2 12/ 3a a v yv y

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 49

2 « 3 2« 2 /2 , 2« 2\ ,

2/l+«a\ 7 2. 2. 7 . /l+a«+a/\

2 , , a 3 3« 3 /, 3« 3\ /l+a«\ 7 _ _,

3 a a a V a a 3 / \ a /

2 . , 7 „. (1 +aa + al\

„| (tf , + £ + c/ + ^/ 3 + /<) - §(« 3 + / 3 ) ( 1 + a « +tt * ) ,

if we denote « 3 j- r by C.

a 2 a'

By the substitution of this value and the corresponding value of
the coefficient of e al , the attraction becomes

AM ?yr"- («-' («■ + - * « ♦ — p + 1 1 )

3 Fa« 2 I v a a y

_ e +ai («i + £ _ ci - 1 - h ^ / 3 + /')

a a

, . 7N ll + a(l + al , 1+aa—al A 1

- (a 3 + /') ( <?- a ' <? a/ ] | .

30. I proceed next to find the value of the term omitted, by taking
the mass of displaced caloric between limits involving B itself. It is
obvious, that we have calculated the attraction of a mass which does
not exist, and shall have to subtract the value which we obtain in
order to get the correct attraction.

Now we have to estimate the resolved part, along the line joining
the centres of the molecules, of the attraction of a mass of fluid of
variable density on the different parts of a solid conceived to occupy
the same space with itself.

If we take any element P of the fluid, and estimate its attraction
on the whole solid, the result will obviously be the attraction of the
solid on this element.

Vol. VII. Part I. G

50 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

Let p be the distance of an element from the centre of the particle
B, and retain the remaining notation of the last problem, then will be
the volume of an elementary annulus

= 2irp 2 dp d6 sin 9,
and the mass attracting this, is

I 3 '

consequently the resolved part of the attraction in the required direc-
tion,

-M£ „ . , ,„ • „ n PA.er« r

y

rrMo 3

= ffjr^-- 2 *P 2 d P de sin OcosO

2ttAMP rr P *e-<" .
= j s / / ' sin 9 cos 9 dO dp.

Now r"- = « 2 + p - 2ap cos 9 ;

.•. rdr = ap sin 9 d9,

and the attraction

ZirAMP

a I 3

ffe- ar p* cos dr dp

= 75 J J e ~ P a — dr d P

al 3 JJ ' Zap r

= - g , f ffe— r P (« 2 + P ~ f) dr d P
ttAMP

at a \a a a I

the limits being r = a + p, and r = a — p

ttAMP r , la* + p 2 . , la-p 1a-p 8\ , A
a V a a 2 a 3 / J

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 51

r i , f«' + /o 2 „„ {(f-Qao + p 2 2a- p 2\

a V a a a / J

2,*APMe- aa ri , [lap a a 1\ ,

+ (T + 5 + ? + ?) e "i

- ! = 4 s^[{e*>j-(?-**5)-e- + 7)-e-j)}'-
M(H)-(^ + 3)*H)-H)M

+ c

_ ZirAMPe-™ rf l + aa /£ _ 2/ 2\ l+a« // _ l\ I a/

0*7" L{ a 2 U a 8+ a 3 j « 3 U a')}*

which must be subtracted from the expression above, in order to give
that part of the total action designated by (3).

31. With respect to the fourth part of the total action, it is evi-
dent that it differs from the expression already obtained on the pre-
vious hypothesis (19) only in sign. Its value is, therefore,

^5^(1 + aa)e— {(l+al)e-° l - (l-al)e« l \,

or in the approximate form

4>irhAe- aa

3 c? a?

G 2

(l+a«)a 2 / 3 .

52 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

32. By collecting all the terms together, we obtain as the total
action of two particles of matter surrounded by atmospheres of re-
pulsive caloric, estimated in the direction of the line joining their
centres, and supposed of an attractive character,

M* IvP qMP

+ 4nrMRd[_ aI (1+a / ) _ c -«(.-0(l +a ^Z7 ) J

2irAMPe~ aa ( ... C ni 1+art
+ ^ \ e ~ W + — + CI+ P + r)

aa I* [a a

- e+° l (a 4 + - - CI -±±^l 3 + I*)

a a

_ («. + P) (Lt££±iJ er-' - 1 + a ;~ a/ «•')}

2,TrAMPe- aa „ ' f/ rt 3/ 3\ . /„ 3/ 3\ ,1

a '

But we have seen (24) that

4irP(l+al)Ae- al ,, 4tt/ j _

^ =M-— Pq;

hence the first line is destroyed by the first term in the second, and
we get

„ AMP2-we- aa r 2e a ' .

S = = — — \l + a(a-l)\

a a 2 L a n

e '"'i^ \ ( a3 3 \ ( . o 1+ ««\ 7 l+"« ,, J

+ ^-|(l +a a).(---j + ^-3-^-)./ + - T -.^ + ZJ

-_|(l + a«).(-- a4 )-(^-3^-)./--— -./. + /«|

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 58

(a 3 + P) (1 + aa + al . 1+aa-al A

— Hi — : — ■" ; — ••")

an expression which involves e~ aa as a factor, and which is conse-
quently of an entirely molecular nature.

The ahove expression may be put into the following form :

S = 5 [2le al

a a L

x , 2e al aV-3 , , , N 8(«-«'+e" J ) 1 , ;

+ (l+a«).{- _ + __(*-'_<>««)-- -A* ^+-(«-' + ««0

a 3 + Z 3

AMP2Tre- aa i, , .. (a? + l 3 a? + 1 3 \\

+ &c.

■ a . //r'o 3 — 3

+

AMP2ire- aa „ ( , , , N /a 3 a 3 -3 , 1 « 3 + f

a«" ' tv y V a / 3 a a/ 3

AMP 2Tre- aa ^ ... , .. /« 3 a 3 a 3 a*J*\ ~»,

AMP2-n-e- aa „ , (e~ al —e al

{p—al pal 1
j-L-+Qah\

54 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

= -33 (1 + ««)

a 0!

|>/p (1+a/e 1-aZe } — i — V

which is a very simple form, and is perfectly general, with the only
exception, that we have omitted all consideration of the caloric displaced
by the material particles between A and B.

33. In order to complete the expression, all that remains to be
done is to find the value of Je or h.

Now h evidently varies as the force on an individual particle of
caloric at the surface of a. material particle.

The expression parallel t;o x for this force is then

This may be divided into two parts, the one that which depends on the
particle of matter at whose surface the force is supposed to act, the other
the united effect of all the other particles. With respect to the latter, it
is easy to observe that the force at the centre of the molecule is
zero, and consequently that at the surface will be the variation of the
whole force by the variation of X through the space I.

Let, therefore, F be the whole force in one direction on a particle

of caloric in the position of the centre of the particle of matter; then

2dF
will , / be the term in question.

34. We have therefore to find

V^e-""(l + aR) = G.

Let the whole mass be intersected by planes at the distances re-
spectively of the particles, and parallel to yz : e the distance between
two consecutive particles : >) the distance of the particle acted on from
the first plane, »? + re is the distance of any plane.

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 55

Let the plane be divided into annuli of which the radius drawn
from the line of intersection with the axis of x is p, then

2tto^p m + re „ -„ _.

a^ijj-«-*(i + «jB)

is the part of G for this annulus ;

.-. part for plane = ?£ C P d P (,, + r e ) e «* i 1 t"-*?

- -r / dB(r, + re) e~ aR i =;— '■

_ 27r(i; + re)<?- C+ re >
e tj + re

— ~JL *-aln+re)

Hence, G = -^<?- a "2 °V

2tt <?-°»

and F = -

H

6 2 (l-e" ae )

" dr, ~ 7(1^ e~ ae ) '
so that the term in question is

e £ (l-C"««)

<r

The other part of//, is iHJ^*-«'L+f^

a e*

If, however, it were required to find the value of h for a particle
not far from the surface of the medium, it would be necessary that

56 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

we should know the law of variation of density at the surface. For
the sake of obtaining the former, let the density be supposed uniform,

Le~ a€
then instead of -5-7= ^ the term will become

e l (1 - e~ ae )

T /»-"« 7? T

e 2 (l-<r") e 2 ^^ «*(l,-*-«« ) {l re h

the particle being distant by pe from the surface:

by substituting this expression, the force of attraction becomes

„ 4>vAMPe- aa (l + aa) ri „ T l—re—'*),, i-n-PP .
lS = - ~^r ~l\ K - L eHl - e -n) ( ~3W q)

~T~ J "

This is a very simple expression for the mutual action of two
particles of matter. As e diminishes, the attractive part of the force
diminishes, so that there is a resistance to the approach of the particles
toAvards each other.

Suppose a particle situated at or near the confines of the medium
to be in equilibrium : then the sum of all expressions similar to the
above, taken throughout the medium must equal zero.

35. I shall only very approximately find the action on a particle
bounding a medium : for it is obvious that in general the force on
it from the surrounding caloric will differ widely from the force on
a particle in the interior of the medium ; the former depending only
on the particles on one side of that in question, the latter depending
on two sets of particles acting in opposite directions, and tending to
counteract each other's efforts. On this account there will in general
be a rapid diminution of density towards the surface of the medium.
The law of this diminution I have attempted to investigate, but from
the circumstance that the resulting equations involve the mixed dif-
ferences of discontinuous functions, I have not hitherto arrived at any
satisfactory conclusion. I shall therefore satisfy myself with finding the
force on a particle bounding a medium, on the supposition that the
medium is homogeneous.

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 57

The expression for the attraction of any particle, is of the form

r .. gfl -»
6 a* •

Now whatever be the form of the bounding surface, it is obvious
that unless the sphere of sensible action be great, it will suffice to
consider it plane and extending to infinity : we shall then have to
estimate the aggregate force on a particle resolved perpendicular to
the bounding plane.

Let the atom under consideration be the centre of a spherical sur-
face to radius a : take an annulus of this surface such that the radius
vector drawn to it makes the angle 9 with the bounding plane:
the area of this annulus = 27ra 2 cos 9 dO,

and the number of particles in it = — — cos 9 d9 ;

hence the attraction on the particle in question resolved perpendicularly
to the bounding plane,

27ra s sin 9 cos 9 d9 e~ aa

2

e

{Ha — m),

and the whole force due to the particles in the hemispherical surface, is

IT Ha _■•' 7T

S " 2

6 6

aa — -. me~ aa

In order to find the whole attraction on the given particle, we
must find the sum of all similar expressions taken through the whole
mass : which is

■n-H , o v irme- a °

= -f^ «<r°'(l + 2*— + «r + ")- e *(i- e -'»)

e . \l-e- ae ) e 2 l-e" ae *
This is the expression which ought to remain constant, whatever
be e, so long as the temperature is so. It is obvious that it varies
directly as H, which involves m' — n'q.
Vol. VII. Pabt I. H

58 PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM.

Moreover the density of the caloric at any point

■wA e~ ae

~ q 6 (1 - <?-<")*

We see, then, that the density of caloric is not a proper measure

of the temperature, although if e be small, the variation of density

will be a proper measure of that of the temperature ; each depending

c~ ae
on — — , or on the density of the material particles, which result I

e

obtained in a popular manner in my Theory of Heat, p. 166, and
found some remarkable consequences to accrue from it.

36. From the expression for the force, it follows directly that
if q increase, which it must do if the temperature be increased in
whatever way that increase be measured, the attractive force diminishes,
but this diminution will also be accompanied by a diminution of the
repulsive force provided a increase, and that too, not by the variation of

the common factor e~ aa - —-^ , but by the diminution of -=7= — - .

a 2 * • (1 — e )

Hence, we perceive that the same series of particles will by an increase
in their mutual distances, exert actions on any one particle just
sufficient to retain it in equilibrium, notwithstanding that the quantity
of caloric has been increased. This fully explains the necessity of ex-
pansion by heat.

37. It was my original intention in drawing up the present Memoir,
to have extended the investigations to a set of combinations of particles,
such as I have supposed to unite in the formation of crystals in my
Theory" of Heat, p. 174, but the subject is so extensive, that I am at
present obliged to postpone it for want of time. I will only make
one remark on the subject, which is this, that if in a binary combi-
nation the lines joining the centres of each pair of particles are
parallel to one another, it is obvious, that the attractive force on a

PROFESSOR KELLAND, ON MOLECULAR EQUILIBRIUM. 59

particle estimated in the direction of this line, will be different from
the force in a plane perpendicular to it, not only in value, but also in
form.

For the supposition of contact between the material particles amounts
to that of exclusion of caloric particles, and consequently we cannot
estimate the action of each particle on every other, as though these
two were the only ones of the system, but must add or subtract from
that action a force due to the caloric displaced at the point of junction ;
and further, the repulsion of the caloric surrounding a particle, must
be diminished in the direction of the line joining the centres, on ac-
count of the quantity displaced by the neighbouring particle. The two
sets of forces will therefore be totally different in form in the two
directions. In that joining the centres of the particles, the variation
of the attraction for a variation of q as well as for a variation of a,
will be very considerable, whereas in a perpendicular direction both
variations will be small. But besides this, it will be seen that the

M* — qMP

term which in Art. 29. completely destroyed g , being

a term arising from the displaced caloric, will not now be sufficient
to destroy it on account of the accompanying particle, consequently
a very small attraction varying inversely as the square of the distance
will remain. This attraction cannot have a sensible value as compared
with the other terms when the distance is small; but when the distance
is finite, the rapid diminution of e~ aa renders the other terms very
much smaller than this, and at a considerable distance this term is the
only one sensible : at such distances then, the force varies inversely as
the square of the distance. Thus all the known laws, as well of at-
traction as of cohesion, are explained by the Newtonian hypothesis.

Queens' College,
March 10, 1838.

Note (a). Let A be the centre of the attracting, B of the attracted particle AB = a,

AQ = p, AC=R the radius of any sphere, V= the volume of B = — angle QAM=<j>.

3

H2

III. On Rolling Curves. By Hamnett Holditch, M.A., Fellow of
Cuius College, and of the Cambridge Philosophical Society.

[Read December 10, 1838.]

In the fifth volume of the Acta Petropolitana, Euler referred to
a class of curves which, when caused to turn round fixed centres,
possessed the property of communicating motion to each other without
friction ; he deduced also their characteristic property, that the point of
contact remains always in the straight line joining their centres: he has
not however followed out the investigation so as to furnish actual forms
of curves, neither has this been done by any other writer that I am
aware of, and consequently no method exists by which such curves
can be obtained. But as they are practically employed in a manner
which I shall proceed to explain, and commonly found by a tentative
process, it appeared worth while to search for forms and rules for their
construction, independently of the analytical interest that may be sup-
posed to attach to such investigation.

Let Anm, Bn i m l (Fig. 1.) be two curves capable of rolling together,
and having their centres of rotation A and B fixed at a distance equal
to the sum of their apsidal distances, Am being a long and Bn t a short
apsidal distance, then if nAm be caused to turn round in the direction
of the arrow, it will press against Bnm t and communicate a rotation to
it. This action will, however, cease when the point m has reached « ;
for beyond that point the radii of mAn will diminish, and its circum-
ference begin to recede from the other curve.

62 Mr HOLDITCH, ON ROLLING CURVES.

No continuous motion of B can therefore be derived from that of
A, if they be continuous curves, unless their outlines be treated like the
pitch lines of ordinary wheels, and be indented with small teeth at
regular distances; these teeth, as in the usual forms, projecting nearly
as far beyond the pitched line or circumference as they extend within
it. If this be done, it will be found that the circumference of A will
retain its hold on that of B in all positions, as well on the receding
as on the advancing sides of the curve. A continuous uniform rotation
of one curve will produce a rotation of the other, not uniform, but
continually varying in its angular velocity, as the ratio of the radius
of A to that of B; this becomes then a commodious contrivance for
converting an equable angular velocity into an unequal one, and is
sometimes so used by Mechanists. Fergusson's well-known Cometarium
was constructed on this principle: it is to be found in use in some
silk machinery, where it is introduced for the purpose of correcting the
unequal action of the common excentric in laying the silk upon the
bobbins; it has also been used by Messrs Bacon and Donkin, in their
printing machinery. I am informed by Professor Willis, who drew my
attention to the subject of these curves, and furnished me with the
above practical information, that the copious collections of Messrs Lanz
and Betancourt, and that of Borgnis, furnish no example of the appli-
cation of rolling curves to the purposes of machinery ; which may there-
fore be considered to have been unknown to them.

When two such curves roll on each other, let r be the distance of
their point of contact from the centre of rotation of the first curve,

and the angle made by r with a fixed radius; then -r— is the tan-
gent of the angle the curve makes with r\ and r t and 0, being corre-
sponding quantities in the second curve, ^— l is the tangent of the

angle it makes with r t , and as r and r t are in the same straight line,
and the curves must have a common tangent at the point of contact,
these two angles must be equal, and

rd9 rd6 i
'-' dr ' ' dr.'

Mb HOLDITCH, ON ROLLING CURVES. 63

Also, if c be the distance of the centres, r + r, = c, and

... dr* + r*d<? = df (l + ^) = drj (l + ^*) - drf + r •*,•.

or the differentials of the lengths of those parts of the curves which
have been in contact are equal, and

, rde rde,
.-. r + r= c, and -*— = -3 — - ,

are equations which contain the analytical conditions of such curves.

We will first consider the case of two equal and similar curves

rolling on each other. Since -j- is some function of r, *%-* must also

be a function of r, let it = fir) ; and as r, and 0, belong to a point in
a similar and equal curve,

••• -^r =/CO; and r, = c - r; .: f(r) =f(c - r),

the solution of which equation is f(r) = <j> (r, c — r); any symmetric
function of r and c — r, and if any form be given to <j> in the

equation —7— = <p(r, c - r), the integration of the latter will give the

equation to a curve having the proposed property. If we suppose it
to have greater and less apsidal distances a and b, which most curves
which can practically be used, must possess ; then, as in revolving the
greater apsidal distance of one must come into contact with the less
apsidal distance of the other, a + b = c ;

Now (a - r) . (r - b) = (a + b) . r - r* — ab = r . (c - r) - ab, is a

dr
dd

symmetric function of r and c — r ; and as at apses -73 = 0, if we

rd6 X(r, c — r) , __. N . . .

assume -7— = , v y , where X(r, c - r) is any symmetric

dr V(a — r).{r — b)

function of r and c — r which is not divisible by \/(a — r) . (r — b),
the curve will be confined between the apsidal distances; and sup-
posing also, that X contains only positive integral powers of r and
c — r, this last equation can always be integrated in finite terms.

64 Mr HOLDITCH, ON ROLLING CURVES.

If r - - = x, any component part of X as

s

=(i-r"(i-n(i-)" + (i-n
=s--r-{(i)"-- i i i -(r-^ }•

consists of even powers of x only, and therefore X will contain no
negative powers of x, and will be of the form

a + b\ 2

X(r, c -r) = k l + k(r- ° L ^-) +

and limiting the investigation, for the sake of simplicity, to the first
two terms,

k t + k.{r- t±Jj

we have dO = . . dr.

ry/{a — r) . (r — o)

To integrate this, let — - — m a, — - — = /3 ;

•. (a - r) . (r - b) = (a + b) . r - r 2 - ab = 2ar - r* - a 4 + /3 s = p - (r - a) 2 ;

/V/3* _ (/• - a) s

, A + £)3*cos*d> , ,
Assume r — a = /3 cos d>, then a0 = ' ^ :r^ • a< P

r r a + p COS

£, + £ . (a + /3 COS — a) 2 -.
a + y3 COS (p ' ™

= '—p: . d<t> + kadcb — kfi cos <bd<b.

a + ft COS (p T T T T

Mr HOLDITCH, ON ROLLING CURVES. 65

Now / r^- (See Professor Peacock's Examples)

'J a + /3 COS <p s r '

= . . tan ' y - —

Va* - F Va* - &

- -4=. tan- fV?. tan <) ,

and tan ? from the equation
2

« + 6 a - i , . , , fa— r

r — — r — = — - — . cos is round to be = V r>

2 2 r r — b

n i i (« + *)*

g*, + *- 5 n ,

9 = t== .tan" 1 V-- V* - r + a£rf> - A/3 sin + C

V«6 « r — b

**.+*. k±^'

2

tan

- \/?- VPI

V«J a r - b

- h y/(a -r).(r - b) + k.(a + b). tan" 1 \J t t^L + Q

2k, + k

r-b

{a + by

or 6 = ' „ 2 .tan- s/\. sj r -^±
y/ab b a — r

- k.\f(a- r).(r - b) - k . (a + *).tan-\A h - , (1),

a — r

where 6 is measured from the smaller apse, is the equation to a class
of curves, which for the present may be called self-rolling curves.

If *, = Vab, and k = 0, = 2. tan- V? . \/ r -^ ;

b a — r

Vol. VII. Part I. I

66 Mr HOLDITCH, ON ROLLING CURVES.

, ab 2ab ,.

and r = n = - rr rr the equation to an

2 v , . , fa + b) + (a - b) . cos ?

a cos 8 - + b sin 2 - v J v J

2 2

ellipse round the focus, which is known to be capable of rolling upon
another equal and similar ellipse.

Hence 9 = ^% . tan" 1 \/j • \Z r -^
Vab b a — r

is the equation to the curve constructed in the ninth section of
Newton's Principia, which is therefore a self-rolling curve.

In the equation found above, if r = b,

n I 2k, k.(a + by . . .J

so that the minor apsidal distances recur, the angular distances between

them being = \—^= + l a + / - k Ja+ b)\ .tt.
Wab 2y/ab K j

If r = a,

and the major apsidal distances recur and bisect the angles between
the minor distances: and if that portion of the curve between two
minor distances, including as they do, a major distance between them,
be called a Lobe, the number of lobes in a revolution

f 2k, k.ia + by . , ,J
Wab 2<Vab )

and in order that the curve may return into itself and so be capable
of successive revolutions, this must be an integer = n ;

Mr HOLDITCH, ON ROLLING CURVES.

, 2& k . (a + by , . ,, 2

and the equation to a self-rolling curve of n lobes is

G = /? + A . (« + b)\ . tan- 1 \Tf . V^
\n J be

'r-b

a — r

- ks/(a-r).(r- b) -k.(a + 6).tan-'V- *,

a — r

(3),

where the constants a, b, n, k may be assumed at pleasure, and for any
value of /•, the corresponding value of 9 will be obtained, and the
curve may then be traced by points.

The different forms the curves may assume will be best illustrated
by an example; thus if a = 10, and 6=1, and k = 114, then if

r = 1,

= 0,

r = 2,

n

+ 14°4,

r = 3,

9 = 118 - 8

+ 11.94,

r = 4,

= 131 ' 8

+ 8.74,

r = 5,

= ii!

+ 5.44,

- a 148.4 nrfl

r = 6, = + 2.74,

w

154.8

r = 7,

9 = + .64,

r = 8,

= .9&>,

r = 9,

0= 167 - 2 1.64,
n

r m 10,

0= 180 .

I 2

68 Mb HOLDITCH, ON ROLLING CURVES.

If n = 1 the curve is. of one lobe, and if also k 2 = it is an ellipse ;
and examples are given when h % = 2, 4, 10 and 20 in figures 2, 3, 4, 5,
C being the fixed centre; if k 2 = — 2, — 4, — 6, — 15, — 20, the repre-
sentations are given in figures 6, 7, 8, 9, 10, in all which figures, only
the upper half of the lobe is drawn, as the lower is similar and equal
to it : and although in some of the figures the radius vector has swept
over more than half a circumference, it has returned so that the semi-
lobe has terminated when 9 — ■*.

If n = 2, 3, 4, &c, curves of 2, 3, 4, &c, lobes may be traced from
the above table, and are readily laid down.

If a = 16.95, and b = 6.95, the following is another table for a great
variety of self-rolling curves:

When r - b

1,

9 = — + 45°.6/5-,

2,

76
9 = — + 43.3&,

M

3,

= ^ + 36.2*,
n

4,

= 1038 + 22 .9*.

5,

. ii4.8 _.;

9 = + 10A,

n

6,

7,

9 = 1M6 11.6*,

N

8,

, = im - ism,

n

9,

. 156.2 _.,
= 16. 5k;

Mb HOLDITCH, ON ROLLING CURVES. 69

and an hour is sufficient to make a table for any assumed apsidal dis-
tances. It will be seen that if k be positive, as k increases, the curves
bulge at the greater apse ; if k be negative and increases, the curves
bulge more and more at the lower apse ; this will afterwards appear
from the consideration of the radius of curvature.

Fig. (25) is an example of a two-lobed curve.

In some cases, as in figures 8, 9, 10, the semilobe commences at the
minor apse by a retrograde motion of the radius vector, and terminates
in such cases by a retrograde motion at the major apse: for let A be
the value of near the smaller apse when r = b + as, and B the value of
ir — 6 near the major apse when r = a — as, as being a very small quan-
tity, then we get from the equation to the curve ;

and therefore if k be positive, A and B are positive ; and if k be nega-

A a

tive, A and B will be both positive, or both negative ; for ^ = t » so that

if a portion of the upper semilobe is below the axis at one apse, there
will also be a portion below at the other apse.

As k increases the curves run into hooks, the points of which have

a tangent passing through the centre, and there can only be two in

/ a + b\ 2
each semilobe determined from the equation k t + k. (r — J = 0, for

d9
at these tangent points — = 0, and this equation has only two roots, the

sum of which is a + b and therefore in rolling they come into contact.

If the value of k t from equation (2) be substituted in this last, the
distances of the tangential points from the centre are

*±* ± \/ _ y^> + E±3 . (j-a - Vby. (5).

70 Mr HOLDITCH, ON ROLLING CURVES.

If k be positive, there are no tangential points unless k is equal to,
or greater than,

4s\/ab

n(a + b). (\/a - \Zbf '

they begin at r m — - — , and as k increases, one moves nearer to, and

the other farther from the centre; and when k is infinite,

a + b y/ a + b . r- /Ts •
■g- ± — (y/a - s/b).

r =

If k be negative, the values of r in equation (5) must be within the

limits of the curve, and therefore there are no tangent points unless

2
k > — jr— , and if k be infinite their distances are the same as

n yv a — y/of

when k is positive : comparing this condition with equation (4) it will
be seen that when k is negative and the curve is retrograde at the apses,
there are always tangent points.

Other forms of self-rolling curves may be found, as

and 9 = A .hyp log r + Bar + (C« 2 - B) . - - ^'. r 1 + — ,

2 3 4

the latter including the logarithmic spiral.

Fig. 22 is a self-rolling curve, where the minor apsidal distance
vanishes, and rolls round the point C in its circumference.

We will now proceed to the consideration of rolling curves when
they are not necessarily similar and equal to each other. If c be the

distance of their centres, and -=— =f(r) be the differential equation of

one of the curves, and r t and 6 / belong to a curve that will roll with
the former, then, since

rdO r t d6, .^. „ ,

^ = i^' and ^>=^ c - r) '

Mr HOLDITCH, ON ROLLING CURVES. 71

it follows from what has been observed before, that

r.d 9 ' .

will be the differential equation of the latter; and any form being given
to f, the integration of these equations will be the equations to a pair
of rolling curves; and for other values of c, other curves may be found,
and so a system formed.

The equation to one of the curves being assumed to be that which
has been found for self-rolling curves, viz.

It + k [r

rdd ' \ 2

dr ' y/( a - r ) . (r - b)
the equation to the other will therefore be

r'de. *, + *-(«-',-nr)

dr, " y/( a - c + r[) . (c - r t - b) '

let c — b = a,, and c — a = b\ ; .*. a,— b,= a — b,
, a + b a + b,

. r M ^'-(H 3 ---)' '.+ '•(-■-'4^'
dr, n/(u t -. r ).{r t -b) Via,- r t ).(r,-b t )

which is of the same form as the differential equation of the assumed
curve, and therefore if n, be the number of its lobes,

,,-g +*.(..+*,)}. <»- V|.VJ£*

- *V(a,- r).(r,-b)-k.(a l +b).tzrr\ V^rf.

a, - r.

72 Mr HOLDITCH, ON ROLLING CURVES.

is the equation to a curve which will roll with the former, the equa-
tions of condition being « — b t = a — b,

,111c, k (a, + 6,)* , . ,. 2

Hence, if h, k t and a — b be any assumed constant quantities, the
values of a and b may be found for n = 1, 2, 3, &c. from the equation

2£, h (a + b)°- t . -,,2

Vah r 2- ^ ' r "»■

by the solution of a cubic equation, as will be easily seen, and the
curves constructed from the equation

9 - (• + *.<« '+ 4] • tan- V?. V^H?
[n ') b a —r

- k s/(a -r).(r - b) - k . (a + b) . tan" 1 \J r -^-,

ct — r

and a system of wheels or curves thus found will roll together in pairs
or in any combinations.

When there are tangential points in one wheel, there will be corre-
sponding ones in all of the same system, and in rolling they will come
into contact with each other; for those of one wheel are found by mak-
ing f{r) = 0, and if a be a root of this equation, c — a, or a will be
a root of the equation f{c - r) — 0, or of f{r) = 0, and .*. a + ft/ = c.

Forms of wheels are readily found from assumed values of & and k t :
or if the dimensions of a pair of wheels be assumed, k and k t may be
found from equation (2) ;

Thus, if n = 1, b = 11 __ . ,,, _,. ., „.

n = 3, b = 5) a = 10 > and ■■", = 1 ^ *+m

rf 1 2 », \Z l) a = 10 ' and •• a ' = 13 ^- Fi ^- ( 12 )-

" " ■ i' ; J I J}« = 10, and ... a 4 - 13}. Fig. (13);
in all which cases each curve is also a self-rolling curve.

Mr HOLDITCH, ON ROLLING CURVES. 73

In this latter example, it will be observed, that the curves are re-
trograde at the apses, which will be the case with all unequal curves
that are made to roll together, if they have the same number of lobes ;

or ' + ' vi/ r y^,.(v^ - \Zb,y - s ~a~b. (Va - \Zby

from equations (2), and if n = « , this expression may be proved to be
negative when a and #,, and therefore b and b /} are unequal: and since

and ~*HVa,-^,r=^.{k, + k.(H^));

therefore, by equations (4), the curves are retrograde at the apses.

If two curves roll one within the other round fixed centres whose
distance is c, then

h h ( a + b y

. rd9 r,de t , .„ rd9 «< + *'\ r - 2 )

r = r + c, and — t— = -~j — * , and it —>— = — , , _ — ,

' - ' rfr dr, dr y/{a-r).(r-b)

be taken for the differential equation of one ;

a + b\

rde t

iil a + by

& t + k.[r, ± c —J

dr, *S( a -r / + c).{r l ±c-b)'
will be that of the other;

let a + c = a, ; and b + c = b, ;

, a + b _ «, + b,

... « _ J) = a 7 - 6 , and — — + c = — ^— ;

and'-*.

*,**-{4M ^ »•.('. -^

<*r, </(«, _ r) . (r, - b) V{a, - r,) . (r, - b) '
Vot. VII. Paet I. K

74 Mr HOLDITCH, ON ROLLING CURVES.

and therefore, the equations will be the same as for those that roll
externally, and the equations of condition are also the same, and con-
sequently all curves (whose equations are of the forms that have been
considered) that roll externally, are also capable of rolling on each other
internally; in the latter case, the major axes come into contact;

a + b a+ b,
for if r, = a,, r = r i ±c = a,±c = a i + — '—t

a — b a + b a - b a + b ,.„ , ,

= 3 t + = — (- — - — = a : and if r = b , r = b.

2 2 2 2 u

The curves in fig. (13) will therefore roll in the positions, figs. (14)
and (15), which are two different attitudes of the curves.

It will be necessary hereafter to know the radii of curvature at
different points of the curve;

Let

rd9

tet-ksnSl

dr " */(u-r).{r-b) \7Y \Zl-s*'

where s is the sine of the angle between the curve and radius vector,
then p m sr, p being the perpendicular upon the tangent ;

dp _ s ds
rdr r dr '

1 X

? .** X* ;

therefore, to find the radius of curvature at the tangent points where
* = 0, and X = 0,

ds Y dX

s*dr X 3 ' dr '

ds_ *_ YdX

dr ~ X s ' dr '

x * - x*+r- r- ■ ds dX

Mr HOLDITCH, ON ROLLING CURVES. 75

and d ^=2k.{r- ~^j =2k.\/^ when X = 0;

also Y — (a — r) . (r — b) = — — [r —J = — + -^ where 1= a — b;

therefore, if R = radius of curvature at the tangent points,

1 dp ds 2k y/ — k t

+ k

R rdr dr / /«'

V *,■

4

which depends only on a — b, and therefore, the radius of curvature is
the same at all tangent points of curves of the same system.

At apses, Y = 0, and s = 1 ;
2ds 1 dY

s>dr X*dr'
l_

as i , , _ . i .„

d> = -2X-*' {a + b - 2r) = „f, ,JV ' lf r = fl '

and = 5 7^ ' if r = *•

Let R a represent the radius of curvature, when r = a ;

«?.? 1

R. r dr a

2

(*, + *i)

Tf ~ h I / 2 \ 2 * ' )"

Hence also the following equations:

1111
/J,, + R b == « + b '

76 Mk holditch, on rolling curves.

1111

R at + R b , a + f;
R a / R b a, + b>

JL _L I I

R a + R hl - a + b, ■

. , 1 s , ds

Also, as y» - +tt»

,1 s ds s ds #

'*• R r + Rr, ~ r + r, '

Ify(^) De the radius of curvature of a curve at a tangent point;
the radii of curvature when r becomes r + h, are

R=f(r + h)=f(r)+f'(r).h,

and, the corresponding radii of curvature of another curve rolling with
the former, are

R, = <i>(r i ±h) = <t>{r)±V{r).h

= <p{r) + <p x (c — r).h;

also, since the radii of curvature of the two curves at the tangent points
are equal

/(r)«*<r,);

and therefore R - R t = + \f x {r) - <f> l {c - r)}.h.

Hence, if R > R t before the tangent points come into contact,
R < R, afterwards, and consequently the curves cross and change their
rolling sides at the tangent points: except h t = 0, when there is a point
of contrary flexure at the tangent points, which then also coincide at
the mean distance.

Mr H0LD1TCH, ON ROLLING CURVES. 77

a + b ds ., ,

If r = — = o. or the curve makes a maximum or minimum

2 dr

angle with the radius vector at the mean distance; and the reciprocal

of the radius of curvature

k.

The area of a wheel may be found: for the area of a lobe is the
integral from = — - to = - of

•«- {*,+ *. (^)". -»•♦}. '(S±* ♦ SeJ; sin *) .**,

, . , a + 6 , , a + J / , N2

which = — - — . * 7r + « . - =-g . (a — o) . 7T ;
2 lb

therefore, the area of a wheel

a + b , , a + b . ..,

= — — — k.rnr + k . -73- .{a - b) . nir,
2 lo

in which expression, if the value of k, be substituted from the equation

2k t , (a + bf i . .. 2
—=*= + k . - — 7== - k . {a + b) = - ,
y/ab 2Vab ' *

the area of a wheel
= - .{a + b) .Vab + ~- . {4, .(a + b)Wa~b - 2. (a + bf + (a + b) .(a- b)'\,

= *- . (a + b) .y/ab + ^ . (a + b) . { (a - by - 2 . (a + V) . (y/a - y/b)*} ,

= ^.(a + b)Val + ^.(a + b).(\/a-v / by.K^ + Vby-2:(a + b)},

(y/a - y/b\ 4

= ^.(a + b).\/ab — Jenir .(a + b) . I j .

78 Mr HOLDITCH, ON ROLLING CURVES.

Those systems of curves where k = 0, have no tangential points ; for

dd = i

dr ~ ^/{ a - r).(r - b) '

and therefore cannot vanish.

If k t = 0, there is always a tangential point in the middle of each
half-lobe.

The former deserve a more particular consideration, as being in
general more simple in form, and admitting of easy and elegant con-
struction: if a„, b n be the major and minor apsidal distances of a wheel
of n lobes, the equations of condition (2) are reduced to «„ — b x = con-
stant = I,

k, 1

and

\/^J B »'

and therefore, a n = - + \/n 2 k i l + - ,

2 V - ( T|:

I / P

and b n = - - + V n*k? + -,

and the equation to a curve of n lobes will then be

9 = - . tan "

r~ p i

V n*k? + - + -

nk, ' V

2n*k
or, r =

r +

I

2

V n*kj

+ 4

I

2

r +

Wn'k;

Z 2 '

+ 4

2 V»^ J + 7 + /.cos0
' 4

Describe therefore a circle whose diameter is /, and draw (fig. 16) a
tangent at any point A, in which take AC = k t and AE = nk t and
draw EG through the centre: then the apsidal distances for a wheel
of n lobes are EG and EF;

Mr holditch, on rolling curves. 79

for EF = EO - FO = - l - + \/n 2 k? + - A = b„,

2 4

and EG = EO+ OG= l - + \/n*k* + - = a„.

C)

Examples : if k* ■= -, and n = 1, 3, 4 &c, the figures (17), (18), (19),
will roll together, or in pairs, and are also self-rolling curves.

The point of contrary flexure, when there is one, is always nearer
to the centre than the mean distance: for if p be the perpendicular
on the tangent from the centre,

rdd _ k t _ p

~fo ~ V(a-r).(r-b) ~ V^?' "* * dp = °'

r =

» 2 - 1 2ab

n°

— — r ; also since (a - bf is positive,

2ab a + b
<,

a + b '2

» s — 1
and — -j — is an improper fraction ;

a + b

'• r < — 7T~
2

Since r — b, which must be positive,

= -5-7 jt . (n*l — 2a),

» . (o + b) v J '

there is no point of contrary flexure, unless n*l - 2a is positive.

The outline of the lobes may be traced without the use of
logarithms by observing, since the equation in this case is

2ab

r =■

a + b + (a - b).cosn9'

80 Mr HOLDITCH, ON ROLLING CURVES.

that if nO = 0, r = J,

4\/2. ab

IT

12' '" «.(2v/2 + VS + 1) + ft. (2^2 - v^ - 1)'

2tt 4aft

/* =

12 ' ~ 2.(a + ft) + (o — ft) Vs'
2\/2.«ft

>7T

r =

12 ' ' « . (i + v'a) + ft . (\/2 - l) '

4nr 4aft

r =

12 ' 3« + ft'

5tt 4\/2.aft

r =

12' " 0.(2^2 + \/3 - 1) + ft. (2\/2 + 1 -VV

6tt 2aft

/• =

12 ' « + ft'

7tt 4\/2.aft

r =

12' " o.(2^/2 + 1 - y§) + ft.(2\/2 + \/3 - 1)'

8tt 4aft

12 ' r " a + 3ft'

9ir 2\/2.aft

r =

12' " «.(V2- 1) + ft.(\/2 + 1)'

IOtt 4>ab

r =

12 ' " 2. (a + ft)- (a-ft)-\/3'

11 7T 4\/2.aft

r =

12 ' " a . (2 -v/2 - ^3 - 1) + ft . (2 y/2 + Vs + 1) '

7r, r = a.

Hence the following rule : Describe the circle whose radius is the
minor distance, and divide it into n equal parts, each of which will
form the base of a lobe ; divide half the base into twelve equal parts,
and draw straight lines from the centre, through the points of division,
respectively equal to the above values: and the curve drawn through
their extremities will be the outline of half a lobe (fig. 20).

Mr HOLDITCH, ON ROLLING CURVES. 81

The distances may also be found practically, by describing an ellipse
whose axis major is «„ + b n , and «„ — b n the distance between its foci ;
then if straight lines be drawn from one of the foci to the ellipse making
equal angles with each other, and the base of the lobe be divided
into as many equal parts as there are equal angles round the focus :
the distances from the centre to the several points of the lobe are
easily shewn to be equal to the elliptic distances ; and may therefore
be set off from them.

The form of a rack, or curve of an infinite number of lobes to
move with the curves derived from the equation

b
r

Wab T 2\/ab I b v a-

- k.y/{a — r).{r - b) - k . (a + b) . tan" 1 \/Lz3L ,

a — r

may be found by making n infinite and a — b = I, where a and b are
also infinite; and this form is that to which the lobes gradually ap-
proach as n increases : if x and y be rectangular co-ordinates of the
rack, x being measured along its base from one of the apses, and
y be perpendicular to the base, x = bO and y = r — b\

= (a*, \/l + Wl . (a I by) . tan- \/\. \fX
[ ' a 2 a ' ) b I —

x

- bks/ly - f - k.{ab + ¥) . tan" 1 Vr^— .

By Maclaurin's Theorem, the expansion of tan -1 V x- 'V r~^ — »
b l — y

of tan -, (l + t) • V i _ as far as the square of t is

-^•^-("♦•"♦'S)-(i-£*S)

Vol. VII. Part I. L

or

82 Mr HOLDITCH, ON ROLLING CURVES.

t
= 2b* + bl + -r , omitting the negative powers of b, as b is infinite;

4

^.V^.tan-VW,-^
2 a a I -

!/

= («&« + bi + 1) .tan- \Zjhj + { b + i) -^y^?;

also b . \/ly — y* + b.(a + b). tan -1 Vj-^j

r

= b^ly-tf + {2b* + bl) . tan" 1 \/X ,

which quantities being substituted in the above equation, we have the
equation to the rack

x = («, + «j . tan- VjL ♦ * . ^ .v^i

from which it appears that each lobe of the rack is composed of four
similar and equal parts.

This equation may also be found from the differential equation

dx

k ' + k \y-i)

which is immediately deducible from

b *

rde *' + *•('•- H")

dr ' y/{ a - r).{r-b)

If & = 0, y = /.sin a -y (fig. 21), which is a rack that will roll with
figures (17), (18), (19).

HAMNETT HOLDITCH.

NOTE ON FRICTION WHEELS.

It was observed, that a rolling continuous curve cannot drive another after
the driving point has reached its maximum distance : if, however, the curves are
discontinuous, and a new driving point shall come into action at the moment the
former driving point shall have reached its maximum distance, a continued revolv-
ing motion without friction, may, under certain circumstances be produced ; and
this will be the case if two wheels be formed of semilobes of the same system, if
clogging of the wheels can be avoided ; for (fig. 23) when the driving point of
A has arrived at G, a new driving point will come into action at B.

The variation of the angular velocity of the wheel driven, supposing that of
the driving wheel to be uniform ; the oblique mechanical action of the driving
wheel near the apses, which at the apses is towards the centre, and the shocks pro-
duced at the change of the driving points, which would however be received at the
flat surfaces, would unfit such wheels for the purpose of moving weights ; it may
still be a question, whether they might not be successfully employed for purposes of
motion.

When the new driving point comes into action, it is necessary that the point
F should clear itself of the point G. The relative motions of the wheels will be
the same if the wheel B be supposed at rest, and the other to move round it ;
and therefore the point F must describe a curve without the wheel B, or the
radius of curvature of the curve described by the point F, immediately after the
change of the driving points, must be less than the radius of curvature at G, sup-
posing the curvature at G to be convex towards the centre of B; in which case,
the wheels will not clog at G, when the driving point is changed.

Let R be the radius of curvature described by F; if the wheel A be sup-
posed to have moved a little, the motion of F will be perpendicular to FC ; and
GH, FH being consecutive normals, FH will be the radius of curvature of the
curve described by F, CD the radius of curvature of the wheel B at the major
apse, and CE that of A at the minor apse.

12

84 Mr HOLDITCH, ON ROLLING CURVES.

CD

Let the small angle BDC*=6; .: z CEA = Q.-~

V.U. *J . CE _ CD

„„„ „ (CD CD\

■■• < FCE = d icE-CF)>

sinC j£ /Ci> CZ>\

or, if a and 6 be the major and minor distances of the wheel B, and R a , R b re-
present the radii of curvature at B and G, and similar quantities with dashes those
of the other wheel, and a - b - I; then

l-R-R a =(l-R).R a .(~-^j;
P.(R bi -R a )

and therefore, R =

l.(R bl -R a ) + R a .R bi

To prevent clogging at G, therefore R < - R b ,

I 1

I i I R a Rb

.-. by equation (6) — — - --<-+-

*/ + *-)

ly b i P(R b -R a )

i

< T +

1 '■(---)

U, Rj

Now — = - +

R„ a f. , JV'

Mi)

and

the curves being considered convex to the centres at the minor apses;

Mk HOLDITCH, ON ROLLING CURVES. 85

1 1 * J A

~r ~ p~ = « + r » and

"■a ■**», a "/

/ 11 1

M" 6 ' --C-8

It may be shewn in the same way, in order that the wheels may not clog at
the point B before the driving point at A comes into action, that

1 1

»(»■♦*$)■ "• ' "(H)'

and as a, b, a /t b, must be positive quantities, both these conditions will be ful-
filled, if

I 1

(*. + *i)

*V l '

I* I

or, *+*._•__, ( 7 ),

which may be called the clearing equation; if the value oi'k t from this be sub-
stituted in the equation

we have finally

2 A, k.(a + bf r 2

-7==+ V ,—T -k.(u + b)--,

Vo6 • n

for determining the radii of a friction wheel of 2 n teeth ; and by giving different
values to n, sets of friction wheels will be found which will not clog theoretically
just before or after the change of the driving teeth : and such wheels will not clog at
other points, unless the depth of the teeth be very great in proportion to the radii
of the wheels, or the curves used for the construction of the teeth be of complicated
forms.

86 Mr HOLDITCH, ON ROLLING CURVES.

An example is given in figure (24), where k = 0, and the clearing equation (7)
becomes 2& 8 =/ s , and equation (8) for determining the radii is therefore

y/zab^nl, and a = -. (l + \/2n 2 + l),

2

6= -.(-1 + V2n 2 +1).

Hence, for a wheel of eight teeth, which is derived from a curve of four lobes,

=3 - s n,if*=i.

= 2-37J

l 4 = 4-771
b m 3-771 '

n = 4, and a = 3 - 37] .^
b

If » = 6, a 7 = 4-77)
6

for a wheel of twelve teeth to turn with the former, and the teeth (or half-lobes)
may be described from rules before given.

The flat sides of the teeth must be a little hollowed out to allow of the free
motion of the points, but these have no connection with the rolling sides.

IV. Note on the Motion of Waves in Canals. By G. Green, Esq. B.A.

of Caius College.

[Read February 18, 1839.]

In a former communication I have endeavoured to apply the or-
dinary Theory of Fluid Motion to determine the law of the propagation
of waves in a rectangular canal, supposing £ the depression of the
actual surface of the fluid helow that of equilibrium very small com-
pared with its depth; the depth 7 as well as the breadth /3 of the
canal being small compared with the length of a wave. For greater
generality, /3 and 7 are supposed to vary very slowly as the hori-
zontal co-ordinate x increases, compared with the rate of the variation
of £, due to the same cause. These suppositions are not always
satisfied in the propagation of the tidal wave, but in many other cases
of propagation of what Mr Russel denominates the "Great Primary
Wave," they are so, and his results will be found to agree very
closely with our theoretical deductions. In fact, in my paper on the
Motion of Waves, it has been shown that the height of a wave
varies as

/8-^7-i.

With regard to the effect of the breadth /3, this is expressly stated
by Mr Russel (Vide Seventh Report of the British Association, p. 425),
and the results given in the tables, p. 494, of the same work, seem
to agree with our formula as well as could be expected, considering
the object of the experiments there detailed.

88 Ma GREEN, ON THE MOTION OF WAVES IN CANALS.

In order to examine more particularly the way in which the
Primary Wave is propagated, let ns resume the formula?,

{lit ; _ e±d w, _/■»-.) .

gdt g \ •> s/gyl

where we have neglected the function f, which relates to the wave
propagated in the direction of x negative.

Suppose, for greater simplicity, that fi and y are constant, the origin
of x being taken at the point where the wave commences when
t = 0. Then we may, without altering in the slightest degree the
nature of our formulae, take the values,

(1) <p = F{x-tVg^),

But for all small oscillations of a fluid, if (a, b, c) are the co-
ordinates of any particle P in its primitive state, that of equilibrium
suppose; (x, y, z) the co-ordinates of P at the end of the time t, and
q> = f<pdt when (x, y, %) are changed into (a, b, c), we have (Vide
Mecanique Analytique, Tome 11. p. 313.)

d<t> , d^> d<&

Applying these general expressions to the formulas (1) we get

$ = r= 'F(a- t y/gy), and x = a -7= F(a - t Vgy)-

Vgy vgy

r

Neglecting (disturbance) 2 , we have

£ = -\A F'{a-ty/gy),

Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 89

and consequently,

supposing for greater simplicity that the origin of the integral is at
a - 0.

Hence the value of x becomes

x = a + - f*dal (a- t \/gy).

Suppose o = length of the wave when t = 0; then £(«) = 0, ex-
cept when a is between the limits and «. If therefore we consider
a point P before the wave has reached it,

J a dat(a-t^gy) = f;da^a)=r;

the whole volume of the fluid which would be required to 'fill the
hollow caused by the depression £ below the surface of equilibrium
when t = 0. Hence we get

, V
x = a H ;

7
x being the horizontal co-ordinate of P, before the wave reaches P.

Also, let x" be the value of this co-ordinate after the wave has
passed completely over P, then

£da%(a - t \/gv) = 0, and x" = a.

If £ were wholly negative, or the wave were elevated above the
surface of equilibrium, we should only have to write - V for V, and
thus

V

x' = a , and x" = a.

7
Vol. VII. Part I. M

90 Mr GREEN, ON THE MOTION OF WAVES IN CANALS.

We see therefore, in this case, that the particles of the fluid by
the transit of the wave are transferred forwards in the direction of the
wave's motion, and permanently deposited at rest in a new place at
some distance from their original position, and that the extent of the
transference is sensibly equal throughout the whole depth. These
waves are called by Mr Russel, positive ones, and this result agrees
with his experiments, Vide p. 423. If however £ were positive, or
the wave wholly depressed, it follows from our formula, that the
transit of the fluid particles would be in the opposite direction. The
experimental investigation of those waves, called by Mr Russel, nega-
tive ones, has not yet been completed, p. 445, and the last result
cannot therefore be compared with experiment.

V

The value — which we have obtained analytically for the extent

7
over which the fluid particles are transferred, suggests a simple phy-
sical reason for the , fact. For previous to the transit of a positive
wave over any particle P, a volume of fluid behind P, and equal to
V, is elevated above the surface of equilibrium. During the transit,
this descends within the surface of equilibrium, and must therefore
force the fluid about P forward through the space

©'

admitting as an experimental fact, that after the transit of the wave
the fluid particles always remain absolutely at rest.

Mr Russel, p. 425, is inclined to infer from his experiments, that
the velocity of the Great Primary Wave is that due to gravity acting
through a height equal to the depth of the centre of gravity of the
transverse section of the channel below the surface of the fluid. When
this section is a triangle of which one side is vertical, as in Channel (H),
p. 443, the ordinary Theory of Fluid Motion may be applied with
extreme facility. For if we take the lowest edge of the horizontal
channel as the axis of x, and the axis of % vertical and directed up-
wards, the general equations for small oscillations in this case become

Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 91

{A) o = g * + ? + *±,

« % ■ p ' dt

we have, also, the conditions

(«) • = jj£ = (when y = 0),

,, w da % *

(») — = -rr = - (when - = cot a),

' v d<f> y y '

dy

a being the angle which the inclined side of the channel makes with
the vertical.

The first of these conditions is due to the vertical side, and the
second to the inclined one, since at these extreme limits the fluid
particles must move along the sides.

Now from what has been shown in our memoir, it is clear that
we may satisfy the equation (2?) and the two conditions just given, by

(c) . <p + £ (tf + *),

<p and <p t being two such functions of x and t only that

It now only remains to satisfy the condition due to the upper
surface. Let therefore

= * - it.,

be the equation of this surface. Then the formula (A) of our paper
before cited gives

o= S~§-§^ (when * =c + £>

M 2

92 Mb GREEN, ON THE MOTION OF WAVES IN CANALS,

or neglecting (disturbance) 8

c being the vertical depth of the fluid in equilibrium.

Also at the upper surface p = 0, therefore by continuing to neglect
(disturbance) 2 (A) gives

= St + 777 (when % = c).
Hence, by eliminating £, we get

which by (c) becomes, when we neglect terms of the order y a and a 8
compared with those retained,

= 2gc<t>, + ^.

Or eliminating (p d by means of (C),

_ d 2 (f> _ gc d\po
df 2 " dx % *

The particular integral of which belonging to the wave that proceeds
in the direction of x positive is

*.=/(*-* \/f) ,
and hence the velocity of propagation of the wave is
(D) ** Vf .

Mr GREEN, ON THE MOTION OF WAVES IN CANALS.

93

Mr Russel gives V -§— as the velocity, but at the same time

remarks, that in consequence of the attraction of the sides of the canal
fixing a portion of the fluid in its lower angle, we ought in employing
any formula to calculate for an effective depth in place of the real one,
p. 442. Instead of adopting this method, let us compare the formula
(D) given by the common Theory of Fluid Motion, with Mr Russel's
experiments. And as in our theory we have considered those waves
only in which the elevation above the surface of equilibrium is very
small compared with the depth c, it will be necessary to select those
waves in which this condition is nearly satisfied. I have therefore taken
from the Table, p. 443, all the waves in which

and have supposed g = 32i

Y
feet: the results are given below.

Observation.

Value of c.

Observed Vel. viz.
feet per second.

Velocity by formula
(D).

4, in.
5,11

6,04
6,05
7,04
7,04
7,04
7,04
7,04

2,19
2,58
2,85
2,88
3,03
3,05
3,04
3,02
3,02

2,313
2,617
2,845
2,847
3,072
3,072
3,072
3,072
3,072

A more perfect agreement with theory than this could scarcely be

expected. Had the formula \Z~i~ = v keen usec *' tne errors w °uld
have been much greater.

94 Mb GREEN, ON THE MOTION OF WAVES IN CANALS.

The theory of the motion of waves in a deep sea, taking the most
simple case, in which the oscillations follow the law of the cycloidal
pendulum, and considering the depth as infinite, is extremely easy, and
may be thus exhibited.

Take the plane (a? as) perpendicular to the ridge of one of the waves
supposed to extend indefinitely in the direction of the axis y, and let
the velocities of the fluid particles be independent of the co-ordinate y.
Then if we conceive the axis % to be directed vertically downwards,
and the plane (xy) to coincide with the surface of the sea in equilibrium,
we have generally,

dx 2 + d%* '

The condition due to the upper surface, found as before, is

dd> d z d>

0= edt-w-

From what precedes, it will be clear that we have now only to
satisfy the second of the general equations in conjunction with the
condition just given. This may be effected most conveniently by

taking

<j> = He'T' sin -^ (v't - x),

by which the general equation is immediately satisfied, and the condition
due to the surface gives

*- T «^. «- , .,

' - V£.

where \ is evidently the length of a wave. Hence, the velocity of
these waves vary as \A, agreeably to what Newton asserts. But the
velocity assigned by the correct theory exceeds Newton's value in the
ratio *s/Hr to \/2, or of 5 to 4 nearly.

Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 95

What immediately precedes is not given as new, but merely on

account of the extreme simplicity of the analysis employed. We shall,

moreover, be able thence to deduce a singular consequence which has
not before been noticed, that I am aware of.

Let {a, b, c) be the co-ordinates of any particle P of the fluid when
in equilibrium. Then, since

27T

5 7T

- HX -VL1 2tt

<b = He \ sm — (v't-x); .-. <t> = — — — e ~ *• cos . (v't - a),
T X 2ttv' X '

and the general formulae (2) give

x = a + -r~ = a — — e *• sin — - (v t - «),
da v X '

d<b H -!?« 2tt , .

% = c + -j-=c-\ r e * cos — - w t — a).

dc v X '

Hence,

and therefore any particle P revolves continually in a circular orbit, of
which the radius is

H «.

round the point which it would occupy in a state of equilibrium. The
radius of this circle, and consequently the agitation of the fluid particles,
decreases very rapidly as the depth c increases, and much more rapidly
for short than long waves, agreeably to observation.

Moreover, the direction of the rotation is such, that in the upper
part of the circle the point P moves in the direction of the motion of
the wave. Hence, as in the propagation of the Great Primary Wave,
the actual motion of the fluid particles is direct where the surface of
the fluid rises above that of equilibrium, and retrograde in the contrary
case.

V. On the Nature of the Molecular Forces which regulate the Constitu-
tion of the Luminiferous Ether. By S. Earnshaw, M.A. of St. John's
College, Cambridge.

[Read March 18, 1839.]

There are already before the world by various authors several
Memoirs, which, collaterally or incidentally, embrace the subject of the
present communication. There is observable in them, however, much
disagreement of results, which seems chiefly to arise from the extreme
length and complexity of the investigations by which those results are
obtained ; to avoid which, as much as possible, their authors are compelled
to adopt means of simplification, which we cannot always be certain a
priori are sufficiently approximative. In the following pages the subject
will be found to be treated in a manner perfectly new and direct, and,
it is hoped also, satisfactory, inasmuch as the analytical operations employed
are brief and simple, involving no principles of a difficult or doubtful
character.

The authors to which I have just alluded have generally adopted, as
a most extensive means of simplification, symmetrical arrangements of
the particles of the ethereal medium. This may be necessary and even
allowable in some cases : but as it has never been shewn that such
arrangements actually do exist in Nature, nor even that they can exist
in Nature, I have been careful to confine myself to the investigation of
properties which are independent of arrangement, or rather, which do
not involve the hypothesis of a peculiar arrangement.
Vol. VII. Part I. N

98 Mr EARNSHAW, ON THE NATURE

I may also remark that the investigations which follow are in other
respects of a very general character. For, in this attempt to discover
the laws of molecular action of the ether, amongst the experimental
properties, assumed as the basis of analytical investigation, are, I believe,
none which are peculiar to the luminiferous ether. I think it probable,
that most terrestrial bodies possess in a greater or less degree of per-
fection the properties here assumed : and consequently, the title of this
paper might have been made more comprehensive. It might, perhaps, not
improperly be, " An Investigation of the Nature of the Molecular Forces,
which regulate the Internal Constitution of Bodies. " This might, however,
be disputed, and therefore in the investigations I have referred only to
the luminiferous ether. Nevertheless, that the reader may more easily
judge what degree of claim the following pages have to that general
character which is here ascribed to them, I shall, in as few words as
possible, introduce a statement of the experimental assumptions, and the
results respectively derived from them.

I. It is assumed that the ether consists of detached particles; each
of which is in a position of equilibrium, and when slightly disturbed
is capable of vibrating in any direction. (Many solid as well as aerial
bodies transmit sound, which is generally supposed to imply the exist-
ence of the same properties in them as are here assumed to be true of
the ether.)

The most curious and perhaps least expected result of this assumption
is, that the molecular forces which regulate the vibrations of the ether do
not vary according to Newton's law of universal gravitation : and it is not
a little remarkable, that a force, whether attractive or repulsive, varying
according to this law, is the only one which cannot possibly actuate the
particles of a vibrating medium.

II. It is next assumed, that the motion of a vibrating particle is
more affected by the influence of the particles which are near to it than
of those which are more remote. (This is certainly true of many other
substances besides the ether.) The result which is sought to be derived
from this assumption is, that the molecular forces which regulate the vibra-

OF MOLECULAR FORCES. 99

tion of the jxirticles are repulsive, and vary according to an inverse
power of the distance greater than 2.

III. It is lastly assumed, that the ether exists (or at least is capable
of existing) as one mass held together by the attraction of its elementary
molecules. This assumption is necessary, in order that the dispersion of
the medium which would naturally result from the repulsive forces which
regulate the vibration of its particles, may be thereby prevented.

The result which is derived from this necessary assumption is, that
each particle exerts (in addition to the repulsive force before mentioned)
an attractive force, which varies according to Newton's Law of universal
gravitation.

By reversing the problem, I have been able to shew, that though
Newton's law is the only one which cannot enable the particles to vibrate,
yet it is the only law of force which can enable them to constitute and
maintain themselves a permanent medium, without endangering, or in any
way affecting their vibrating or luminiferous property.

I have on these grounds not hesitated to express my opinion, that
the particles of the luminiferous ether are each endued with two forces of
distinct characters and uses; one attractive, to preserve themselves a per-
manent medium, varying inversely as the square of the distance; and the
other repulsive, to which is due their luminiferous property, varying in a
higher inverse ratio of the distance than the square.

A SYSTEM OF DETACHED PARTICLES.

1. If V denote the sum of the quotients formed by dividing each
attracting body by its distance from the attracted body ; then V = C is
the equation of a surface at any point of which if the attracted body
be placed, it will begin to move in the direction of a normal.

N2

100 Mh earnshaw, on the nature

For, let f, g, h be the co-ordinates of the attracted body ; F, G,
H the attractions of the whole system upon it parallel to the co-ordi-
nate axes, then

F = d f V,G = d g V, H = d h V.

But the equation of the tangent plane at that point of the pro-
posed surface where the attracted body is placed, satisfies the differen-
tial equation,

= d f F.df + d g V.dg + d k V.dh\

:. 0= F.df+G.dg +H.dh.

This equation shews that the resolved part of the attractive force
is zero, in the direction of the tangent plane ; and therefore the whole
attraction is in the direction of the normal.

2. For the sake of brevity, I shall denominate the surface V=-C,
the parametric surface passing through the point f, g, h.

Different points in space may have corresponding different parametric
surfaces; any one may be found by assigning the proper value to C.
Their equations differ only in the value of the constant C, which,
for this reason, I shall call the parameter.

If any parametric surface pass through an attracting particle, its
parameter will be infinite, because at that point V is infinite; in
which case the proposition will fail. The proposition is true of re-
pulsive forces, or if some of the particles exert repulsive and some
attractive forces ; but when the forces are all attractive, V can neither
be evanescent nor negative: since, however, it is infinite when the
attracted particle touches any one of the attracting particles, and is not
infinite in other positions, there must be some intermediate positions
which make V a minimum, and there may be positions in which V is
a maximum.

3. The parametric surfaces which pass through points indefinitely
near to a point of neutral attraction, are in general similar concentric

OF MOLECULAR FORCES. 101

hyperboloids of one and two sheets, the common centre of which is the
point of neutral attraction. Certain points, however, have the asymptotic
surface for their characteristic surface.

Let fgh be the co-ordinates of K, the point of neutral attraction,
and f + x, g + y, h + «, the co-ordinates of P, a point very near to
K. Let the value of V at K be C, and at P, C. Then the equa-
tions to the respective surfaces are

C = V, and C = V ;
where V is the same function of / + x, g + y, k + « that V is of f

.: C= V + d f V.x + d a V.y + d h V.x + d}V .^ + d 2 g V .%

as 2
+ d\V .- + d f d g V ,xy + dfd h V .x% + d g d h V ,y% + &c.
SB

But because K is a point of neutral attraction, d f V = 0, d g V = 0,
d h V=0, and

.-. 2(C- C) = d}F.x* + dlF.y* + dlV.%* + 2d,d g U.xy + &c.

This, neglecting terms above the second order, being the general
equation of surfaces of the second order which have a centre, by
transposing the co-ordinate axes so as to coincide with the principal
axes of the surface, the terms containing xy, y%, x» will disappear,
leaving only

2(C- C) = d* f V.x* + d g V.tf + d\V.%\

which for indefinitely small values of x, y, % may be regarded as the
equation of the parametric surface. It must be remembered that the
coefficients of a?, if, ss s are subject to the condition,

= d}r+d g F+dlV;

and because at least one of these coefficients will be negative, and one
positive, the equation is that of an hyperboloid.

102 Mr EARNSHAW, ON THE NATURE

4. That parametric surface which contains a point of neutral at-
traction will be a cone, which is asymptotic to all the hyperbolic para-
metric surfaces belonging to the other points.

For, the parameter of the surface passing through K, the point of
neutral attraction is C", and therefore the equation of it is

= d}V. a? + d 2 g V. f + d 2 h V.z\

which is the asymptote of the surfaces included in the equation,

2(C- C) = d)V.x? + d* g V.f + d\V.%\

6. If the position of equilibrium be such, that only one of the
quantities d) V, d 2 g V, d\V is negative, as for instance, d)V; then the
axis of the asymptotic cone will coincide with the axis of x ; and all
points within this cone will have hyperboloids of one sheet for their
parametric surfaces, and their parameters will be less than C". The
points without this cone will have hyperboloids of two sheets for their
parametric surfaces, and their parameters will be greater than C

If the position of equilibrium be such, that two of the quantities
d} V, cCgV, d\V are negative, as for instance, d\ V and d\ V, the axis
of the asymptotic cone and the parametric surfaces will be as in the
last case; but the parameters of points within the cone will be greater
than C", and of points without it, less than C.

7. If the molecular forces are all repulsive, then the sign of V
will be changed : but the pai-ametric surfaces will be hyperboloids, as
before.

8. If the position of equilibrium be such, that d) V = 0, d g V = 0,
and d\V = 0, then d f V, d g V, d h V, i.e. the attractions F, G, H,
being also evanescent, the particle is unattracted in every direction, at
least for small displacements from the position of equilibrium. An
example of this is afforded in the case of a particle placed within a
spherical or ellipsoidal surface, composed of attracting or repelling par-

OF MOLECULAR FORCES. 103

tides. If the position of equilibrium be such, that one of them, as
d) V, is evanescent, then, F being evanescent also, for small displace-
ments parallel to the axis of at, the particle will be unattracted. An
example of this is afforded in the case of a particle placed within a
hollow elliptic or circular cylinder of indefinite length. The displace-
ment of particles placed in such positions as those here considered
would not bring into action any forces of restoration ; on which account
the particles would not vibrate. It is evident, therefore, that the phe-
nomena of light and sound are not due to the motions of particles
placed in such positions : and as the purpose of this paper is to ex-
amine the constitution of media supposed to be capable of transmitting
light, a phenomenon due to vibration, we shall, in what follows, always
suppose that none of the quantities d} V, d 2 g V, d\ V are evanescent :
under which supposition also, they cannot be equal, since their sum
= 0.

9. Since the force which urges a displaced particle acts in the
direction of a normal to the parametric surface in which the particle is
at any moment situated, there are in general only three directions in
which a particle can be displaced, so that the force called into play
may act in the direction of the displacement. These directions coincide
with the principal axes of the parametric hyperboloids. The exceptions
to this are, when the constitution of the system is such, that two of
the three quantities, (P f V, d\ V, d\ V, are equal ; in which case the asymp-
totic cone has a circular base, and the exterior parametric hyperboloid
becomes the hyperboloid of revolution of one sheet : and since the normals
to this surface, corresponding to points in that principal section which
is perpendicular to the axis of revolution, all pass through the centre,
the force of restitution will always act in the line of displacement, when
the particle is disturbed in any direction in this plane. This is the only
exception.

10. It is very important to remark, that since the parametric sur-
faces cannot be spherical in any case, the constitution of a medium,
composed of detached attractive particles, can never be such that the
force of restitution called into play by a disturbance in any direction

104 Mr EARNSHAW ON THE NATURE

shall act in the line of displacement. Hence those media which are
distinguished as uncrystallked, cannot consist of detached particles which
either attract or repel each other, with forces varying inversely as the
square of the distance; because it is assumed as a characteristic pro-
perty of such media, that the forces of restitution act always in the
direction of displacement.

11. To find the force of restitution, when a particle is slightly dis-
turbed from its position of equilibrium.

Let F, G', H' be the resolved parts of the force of restitution
parallel to the co-ordinate axis upon the particle at P; then F' is the
same function oi f + x, g + y, h + z, that F is off, g, h, and therefore

F' = F + d f F.x + d 9 F.y + d h F.x + ...

or, F' = d f V+ d}V.x + d f d g V .y + d f d h V .% + ...

= d}V.x + terms involving a? 2 , y 2 , %-, xy, &c.

because d f V = 0, d f d g V = 0, d f d h V = 0. (Art. 4).

Similarly, G' = d\ V . y + ...

and H' = d\V .% + ...

Hence, if the system consisted of fixed particles, the particle P
only being moveable, the equations for P's motion would be

dr t x = d) V . x \
tt t y = dlV.y

d\z = dlV.%\

very nearly.

It is remarkable, that — — + — + — = 0.

x y x

12. From this investigation it appears, that the force of restitution
parallel to any one co-ordinate axis depends only upon its displacement

OF MOLECULAR FORCES. 105

parallel to the same axis. We may therefore consider the effect of each
component of the displacement separately.

It appears from the equations just obtained, that — d)V, -— (P g V,
— d\V are the absolute forces of restitution.

Since one at least of the quantities d) V, d] V, d\ V is negative, and
one at least positive, there will be at least one principal axis parallel
to which a disturbed particle can vibrate, and at least one parallel to
which a disturbed particle cannot vibrate. Suppose for instance, that
d}V is positive and d\V negative, then the first equation dfx = d}V. x
takes the form

dfx — a 2 X,

the integral of which is

x = CV' + C"e- at ;

a result which shews that x must increase continually with t. The
motion in this direction will therefore be one of translation.

But for that part of the displacement which is parallel to the axis
of %, the equation of motion is

d?% m — 7 2 s.

The integral of which is

% = A cos (7/ + B),
which denotes vibration.

13. If the constitution, or arrangement of the particles, of the
medium is such that d g V is positive, the motion parallel to y will be
one of translation; and consequently there will only be one line in
which a particle can be displaced, so that its motion may be vibratory.

14. If the constitution of the medium be such that d* g V is negative,
the motion parallel to y will be vibratory; and therefore if the particle
be displaced in any direction in the plane y%, it will continue to vibrate
in that plane, describing an elliptic orbit.

Vol. VII. Part I. O

106 Mr EARNSHAW, ON THE NATURE

15. It appears then that at the most, the equilibrium can only be
stable in one plane; and that the medium may be so constituted that
the equilibrium shall be stable only in one line. The character of in-
stability, which in the preceding articles we have shewn necessarily
attaches to a medium constituted of particles placed at finite intervals,

and attracting each other with forces varying as -=p, cannot be removed

by supposing the particles to repel each other with forces varying ac-
cording to the same law. The equation d} V + d*K+ d\ V = 0, frOm
which the instability arises, holds equally for attraction and repulsion.

It may be observed also that the instability cannot be removed by
arrangement; for though the values of d)V, d^V, d\V depend upon the
arrangement of the particles, the fact that one at least must be posi-
tive and one negative depends only upon the equation d}V+d\V + dlf r =0,
which is true for every arrangement. And consequently, whether the
particles be arranged in cubical forms, or in any other manner, there
will always exist a direction of instability.

It is therefore certain, that the medium in which luminiferous waves
are transmitted to our eyes is not constituted of such particles. The
coincidence of numerical results, derived from the hypothesis of a
medium of such particles, with experiment, only shews that numeri-
cal results are no certain test of theory, when limited to a few cases
only.

16. It has been noticed, that the instability of a system depends
upon the equation d) V + d% V + d\ V = 0. With the ordinary law of
attraction it always holds good. If, however, the force of molecular

attraction be assumed to vary as -j^, and we write

V for 2 l!-4 ,
n — 1

we shall find

d}V+ d)V + dlV= (n - 2) 2 (A) (i).

OF MOLECULAR FORCES. 107

By an investigation precisely similar to that in Art. 11, we find
F' = d}V.x + ..

G' = d* g V.y + ...

H'-dlF % + ..

Now, since 2 [-^r,] is necessarily positive, one at least of the quan-
tities, d}V, d 2 g V t d%V, in equation (1) is necessai-ily positive for all
values of n equal to or greater than 2 ; and consequently, one at
least of the equations (2) must necessarily denote translation. And
this is true whatever be the arrangement of the particles.

But when n is less than 2 the right-hand member of (1) is
negative, in which case it is possible that all the equations (2) may
denote vibration.

Hence, if the luminiferous ether consist of detached particles which

attract each other with forces varying as yr , n must be less than 2.

In a similar manner it may be shewn, that if the ether consist
of repulsive particles, n must be greater than 2.

It must be remarked, however, that although these conditions with
regard to the value of n should be satisfied by the law of attraction
of the particles, yet their arrangement must be such as shall make
dyV, d\V, d\V all negative for every particle in the system, other-
wise it will be unstable and incapable of transmitting light.

17. If the medium be of the kind denominated uncrystallized, the
vibration of a particle in any direction must be completed in the same
time, in which case the arrangement must be such as simultaneously
to satisfy the equations,

d}V=dlV=dlV=^ s(2!L) f
n being less than 2.

o2

108 Mr EARNSHAW, ON THE NATURE

We shall arrive at the same result if we consider an un crystallized
medium to be such that the force of restitution acts always in the
line of displacement; for in this case the parametric surface, the ge-
neral equation of which is

2(» - 1) (C- C) = d}V.m? + d^V.f + d\V .*" (Art. 3).

must be spherical ; which requires that

d}V = d>V=dtV.

18. It can be easily shewn that n must be greater than unity.

For the number of particles at the distance r from the attracted
particle is proportional to r 2 , and therefore

n - 2 m r 2

2 oc 2 ,

- 1

oc 2

r n-\ '

hence, unless n be greater than unity, the effect of the more distant
parts of the medium upon the value of — - — 2— ^ will be greater
than the effect of the adjacent particles. Now the time of vibration
of a particle depends on the value of dj V, or — - — . 2 ——^ ; and there-
fore unless n be greater than unity, the parts of the medium which
are more remote will exert a greater influence upon the time of vi-
bration than those exert which are near. Now, Optical phenomena
seem to indicate that the adjacent particles exercise most influence ;
and therefore n must be greater than 1.

19- It is probably not conformable to the simplicity of Nature,
that n should be fractional ; we have shewn that it must be greater
than 1 and cannot be equal to 2, consequently n is greater than 2.

This result is important, as we are enabled to infer from it imme-
diately, by the aid of (16), that

OF MOLECULAR FORCES. 109

If the ethereal medium consist of detached particles, the action of
which on each other is 'proportional to a power of the distance, that
power must be greater than 2, and the force must be repulsive.

I have pleasure in remarking, that this result so far as it goes,
coincides exactly with that which M. Cauchy has obtained in his
" Me"moire sur la dispersion de la lumiere," page 191, where from his
investigations he infers respecting the mutual action of two molecules
of ether, "que, dans le voisinage du contact, cette action soil repulsive
et reciproquement proportionelle au bi-carre de la distance."

20. If the particles of ether exert a repujsive action upon each
other, as we have just shewn must be the case, they will naturally
endeavour to disperse themselves through all space, and form a medium
coextensive with the boundaries of the universe. Here then a for-
midable difficulty presents itself to our notice. If the medium be of
finite dimensions it must be enclosed in an envelope, capable of re-
straining the expansive energy of the whole mass of particles. The
more extensive the medium the greater must be the strength of the
envelope. Is it probable that the constitution of the Universe is such
as to require that the whole should be enclosed in a huge vessel of
inconceivable strength? This objection would in my opinion be fatal
to the hypothesis of a system of detached particles, were it not for the
following considerations.

Upon examining the preceding articles, it" will be seen that the
luminiferous ether must be such that d}V, d 2 g V, d\V are all equal and
negative. Now the properties of these quantities will not be in the
least affected, if we assume that the particles exert attractive forces as
well as repulsive forces, providing the attractive forces are proportional

to -=p. For let us suppose that

V =

where n and m are respectively the attractive and repulsive forces exerted
by the same particle at the distance unity.

110 Mr EARNSHAW, ON THE NATURE

Then as in (16), we have

d}V + dlV+dlV= - (n - 2) 2 (J^) ;

equations which do not contain the quantity /d.

I think it therefore not improbable, that each particle of the lumi-
niferous ether exerts two forces, one attractive . and varying reciprocally
as the square of the distance; and the other repulsive and varying
inversely in a higher ratio than the square; at any rate this supposi-
tion does away with the necessity of the envelope mentioned at the
beginning of this article.

21. Let us now generalize the problem, and inquire for what laws
of molecular force vibration is possible in the particles of ether.

Let tyr be the law of molecular force; and assume V = — 2 {mf r <f>r) ;

.-. d)V+dlV+dlV=--2 L(^ + 0V)},
(p'r for brevity denoting d r (pr.

Now one condition to be fulfilled is, that d)V+djV+ d\ V= a nega-
tive quantity, and consequently the law of force must be such that

2 d>r

— *— + <p'r = a positive quantity ;

for all values of r from r = the distance between two neighbouring
particles, to r = oo ; let ^r be any function of r which is positive be-
tween these limits, then

-f- +<pr = ^r;

.'. %r<pr + r*<p'r = r^-^r,

.-. r*<pr = C+ t(f*+r),

OF MOLECULAR FORCES. Ill

This formula contains every possible law of force : the first term
shews the propriety of what we have done in the last article, and further
proves, that an attractive molecular force varying inversely as the square
of the distance is the only force which possesses the properties requisite
for removing the difficulty there stated; or that at any rate it is the
simplest and best adapted for that purpose.

Further; for a reason analogous to that assigned in (18),

r 2 1 — — + <j)'r) , or r^-^r

^ ............ . ■

must be a function of r, which decreases as r increases, and vanishes

when r is infinite. Hence, if %(r) be any function of r which is

positive between the least and greatest limits of r for the whole medium,

and which decreases as r increases and vanishes when r is infinite, then

C 1

<t>(r) = ^ + ^f rX (r).

Every possible law of force is included in this formula; but the
converse is not necessarily true, viz. that every law of force included in
this formula is possible.

There may be other conditions to be satisfied, either as to the form
of the arrangement of the particles, or as to their distance from each
other, or as to the possibility of the medium existing in a state of
finite extension, or as to other circumstances unknown to us at present
which may perhaps exclude all the forms but one; which one would
in that case be the actual law in the luminiferous ether. Or there may
be peculiarities in the vibrations which constitute the waves of light
(such as their transversality) which will hereafter enable us to determine
the required law of mutual action of the particles.

22. Whatever be the law of molecular force of the luminiferous
ether, each particle is placed in such a position when in equilibrium,
that the value of V for that particle is a maximum.

Let us employ the notation of (21): then V=- 2(mf t <pr), and

112 Mb EARNSHAW, ON MOLECULAR FORCES.

d}V + d g V + d\V = - il* {££ + 0>)}

= — 2(/»\J/-r),

and every one of the quantities d}V, d\V, d\V is negative, whether
the particle of ether (the state of which we are investigating) be within
a crystallized body, or in vacuo, or in an uncrystallized body.

In order that V may be a maximum, we must have fulfilled the
following conditions, viz.

d f V=0, d g V=0, d>V=0 (1),

d}V, d g V, d\V all negative (2),

and

d}V.d g V>(d s d g vy
div.d\v>(d g d h vy
d}V.d\v>(d f d K vy

(3).

The three conditions marked (1) are fulfilled, because the particle
is in equilibrium by hypothesis ; we have shewn above that the three
conditions (2) are fulfilled, otherwise the medium could not be lumini-
ferous, i. e. its particles could not vibrate in any direction ; and the
last three conditions marked (3) are fulfilled, because the directions of
the co-ordinate axes have been taken, such that d f d g V =■ 0, d g d h V = 0,
and d f d h V = 0. Consequently V is a maximum.

S. EARNSHAW.

.

VI. Supplement to a Memoir on the Reflexion and Refraction of Light.
By G. Green, Esq. B.A. of Caius College.

[Read May 6, 1839.]

In a paper which the Society did me the honour to publish some time
ago, I endeavoured to determine the laws of Reflexion and Refraction
of a plane wave at the surface of separation of two elastic media, sup-
posing this surface perfectly plane, and both media to terminate there
abruptly : neglecting also all extraneous forces, whether due to the action
of the solid particles of transparent bodies on the elastic medium, which
is supposed to pervade their interstices, or to extraneous pressures. I am
inclined to think that in the case of non-crystallized bodies the latter
cause would not alter the form of our results in the slightest degree ;
and possibly there would be some difficulty in submitting the effects
of the former to calculation. Moreover, should the radius of the sphere
of sensible action of the molecular forces bear any finite ratio to X, the
length of a wave of light, as some philosophers have supposed, in order
to explain the phenomena of dispersion, instead of an abrupt termination
of our two media we should have a continuous though rapid change of
state of the ethereal medium in the immediate vicinity of their surface of
separation. And I have here endeavoured to shew, by probable reasoning,
that the effect of such a change would be to diminish greatly the quan-
tity of light reflected at the polarizing angle, even for highly refracting
substances: supposing the light polarized perpendicular to the plane of
incidence. The same reasoning would go to prove that in this case the
quantity of the reflected light would depend greatly on minute changes
in the state of the reflecting surface. I have on the present occasion
Vol. VII. Part I. P

114

Mr GREEN'S SUPPLEMENT TO A MEMOIR ON

merely noticed, but not insisted upon, these inferences, feeling persuaded
that in researches like the present, little confidence is due to such con-
sequences as are not supported by a rigorous analysis.

The principal object of this supplement has been to put the equations
due to the surface of junction of two media, and belonging to light
polarized perpendicular to the plane of incidence, under a more simple
form. The resulting expressions have here been made to depend on
those before given in our paper on Sound, and thus the determination
of the intensities of the reflected and refracted waves becomes in every
case a matter of extreme facility. As an example of the use of the
new formulae, the intensities of the refracted waves have been de-
termined for both kinds of light : the consideration of which waves had
inadvertently been omitted in a former communication.

Perhaps I may be permitted on the present occasion to state, that
though I feel great confidence in the truth of the fundamental principle
on which our reasonings concerning the vibrations of elastic media have
been based, the same degree of confidence is by no means extended to
those adventitious suppositions which have been introduced for the sake
of simplifying the analysis.

Let us here resume the equations of the paper before mentioned,
namely,

dcp d\j/ _ d<p t d^f, "
dx dy dx dy

(17)

d(j>
dy

djf_
dx

dy

dx

i (when x = 0).

d*(j> d'cp,
g*dt* g ;d?

y*dt* yfdt 2

where u and v, the disturbances in the upper medium parallel to the
axes x and y, are given by

THE REFLEXION AND REFRACTION OF LIGHT. 115

_ d<p d\\,
dx dy '

_ d<f> d\js
dy dx '

u t and v, the disturbances in the lower medium being expressed by
similar formulae in rf> t and ^.

The two last equations of (17) give, since

x = £•= X

<p' = M 2 $,', y = m 2 ^/ ;

<p and <p t being accented for a moment to distinguish between the par-
ticular values belonging to the plane (yz) and their more general values

(j> = e hx <f> and <£, - e" 6 ^/.

The correctness of these values will be evident on referring to the Memoir,
formulae (20), (21), and recollecting that

b = a' = «;.
Hence the first equation gives, since x = 0,

• • *' " " bWTT) ~dy~ ' ana + btf + 1) dy '

Also the second equation may be written,

dj, _d±,_d$__d(ti . 0* 2 - 1)' dfy,
dx dx " dy dy b (m 2 + 1) dy 2 '

And since we may differentiate or integrate the equations (17) relative
to any variable except x, we get for the conditions requisite to com-
plete the determination of ^ and >//,,

p2

116 Mb GREEN's SUPPLEMENT TO A MEMOIR ON

(29) dJ L _d^_ (»* - If gfr I (when a? = 0).

Or neglecting the term which is insensible except for highly refracting
substances,

( 3 °) tf^ _ A^ ( wh en * = 0),

* where m = — is the index of refraction.

These equations belong to light polarized in a plane perpendicular
to that of incidence, and as <p and <p t are insensible at sensible distances
from the surface of junction of the two media, we have, except in the
immediate vicinity of this surface,

_ ety
du

V = — -TV" .

dx

* Though these equations have been obtained on the supposition that the vibrations
of the media follow the law of the cycloidal pendulum, yet as (b) has disappeared, they
are equally applicable for all plane waves whatever.

In fact, instead of using the value

yp- / = a t sin (a t x + by + ct),

and corresponding values of the other quantities, we might have taken the infinite series

yf/ t = Sa ( sin n (a t x + by + ct),

where a and n may have any series of values at will. But the last expression is the
equivalent of an arbitrary function of

a t x + by + ct.

Or the same equations might have been immediately obtained from (17), without in-
troducing this consideration. The method in the text has been employed for the sake of
the intermediate result (29), of which we shall afterwards make use.

THE REFLEXION AND REFRACTION OF LIGHT. 117

When light is polarized in the plane of incidence, the conditions
at the surface of junction have been shewn to be

w — w t

(32) dy^ ' dw^ \ (when x = 0).

dx dx

Since in these conditions we may differentiate or integrate relative
to any of the independent variables except x, we see that the expressions
(30) and (32) are reduced to a form equivalent to that marked (A) in
our paper on Sound ; and the general equations in >// and w being the
same, we may immediately obtain the intensity of the reflected or re-
fracted waves, by merely writing in the simple formulae contained in
that paper,

A = 1 and A, = 1 for light polarized in the plane of incidence;

or A = -5 and A, = — 5 for light polarized perpendicular to the plane of

7 y ' • -a

incidence.

As an example, we will here deduce the intensity of the refracted
wave for both kinds of light.

Representing, therefore, the parts of w and w, due to the distur-
bances in the Incident Reflected and Refracted waves by

f(ax + by + ct), F(- ax + by + ct), and f^ap + by + ct)

respectively, and resuming the first of our expressions (7) in the paper
on Sound, viz. —

f = * I A + a ) S"
we get for light polarized in the plane of incidence, where A = A, = 1,

2 cos 9 sin 0,

/

1

2

+

5
a

2

f~

1

COte,
+ COt0

sin (0, + 0) '
which agrees with the value given in Airy's Tracts, p. 356.

118 Mr GREEN'S SUPPLEMENT TO A MEMOIR ON

For light polarized perpendicular to the plane of incidence, we
have A = — and A = — ;. If, therefore, we here represent the parts
of f and yff t due to the same disturbances by / F and /, we get

f t 2 sin 9, cos 9 2

f'~^f_ cot 9, m sin 9 cos 9, ' cos 9 sin 9 '

yf + cot 9 cos 0, sin 9,

Also, if B be the disturbance of the incident wave in its own
plane, and D t the like disturbance in the refracted wave, we have by
first equation of (31),

B sin 9 = u = Y- = bf {ax + by + ct),

and B, sin 0, = w, = ^' = bf, (ax + by + ct),
retaining in >//• the part due to the incident wave only.
Thus by writing the characteristics merely,
Si sin fl /' _. cosg

B sin 0, f cos 9 t ' cos 9 sin 9

cos 9, sin 9,

cos 9 sin 9

cos0 I, ~ cos0. sin ft
Hi +

cos0, | cos sin 9

cos9 f tan (0, - 0)

COS0

1 d * tan (0,-0) 1
»V , tan(f+ft)J»

cos 9 t sin 7
which agrees with the formula in use. (Vide Airy's Tracts, p. 358)<

In our preceding paper, the two media have been supposed to ter-
minate abruptly at their surface of junction, which would not be true
of the luminiferous ether, unless the radius of the sphere of sensible
action of the molecular forces was exceedingly small compared with \,
the length of a wave of light.

THE REFLEXION AND REFRACTION OP LIGHT. 119

In order, therefore, to form an estimate of the effect which would
be produced by a continuous though rapid change of state of the
ethereal medium in the immediate vicinity of the surface of junction,
we will resume the conditions (29), which belong to light polarized in
a plane perpendicular to that of Reflexion, viz.

and instead of supposing the index of refraction to change suddenly
from to /a, we will conceive it to pass through the regular series of
gradations,

MO. Ml! l"2j M3 fi„;

t being the common thickness of each of these successive media.

Then it is clear we should have to replace the last system by
m to = to, and $ - % - £^ £ C . T ),

to = to, <*%-%- J^fj Sf c* = •*

*->*-«--**• "* "3F" ■ "3* " m!-, « + rf-0 6 ^ {•-(■-i.-OI.

But it is evident from the form of the equations on the right
side of system (33), that the total effect due to the last terms of
their second members will be far less when n is great, than that due

120 Mr GREEN's SUPPLEMENT TO A MEMOIR, &c.

to the corresponding term in the second equation of system (29)*.
If, therefore, we reject these second terms, and conceive the common
interval r- so small that the result due to the first terms may not
differ very sensibly from that which would be produced by a single
refraction, we should have to replace the system (29) by (30), and the
intensity of the reflected wave would then agree with the law assigned
by Fresnel. In virtue of this law, however highly refracting any
substance may be, homogeneous light will always be completely po-
larized at a certain angle of incidence ; and Sir David Brewster states
that this is the case with diamond at the proper angle. But the phe-
nomena observed by Professor Airy appear to him entirely inconsistent
with this result (Vide Camb. Phil. Trans., Vol. iv. p. 423.) ; what im-
mediately precedes seems to render it probable that considerable dif-
ferences in this respect may be due to slight changes in the reflecting
surface.

* In fact, in the system (33) each of the last terms will, in consequence of the factors
(/*,* - 1'*)*' &c be quantities of the order -5 compared with the last term of (29')» and

as their number is only n, their joint effect will be a quantity of the order - compared
with that of the term just mentioned.

C«u»b Phi] Soo Vol. VII. H. 1

Mrtiitlii t- I'aimir l.ifheq ('amindq

Camb Phil.Soc Vol. VII T\ X.

Urteal/r i Tabrur. lukeg Csmiridyt

ciu»b. riui.soc voi vn.ri.3-

Ifctealfe k TeUtntr. Lithof CtunJhrKUft

TRANSACTIONS

OP THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

Volume VII. Part II.

CAMBRIDGE:

PRINTED AT THE PITT PRESS;

AMD SOLD BY

JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J. DEIGHTON; AND T. STEVENSON, CAMBRIDGE.

MDCCC.XLI.

VII. On the Propagation of Light in Crystallized Media.
By G. Green, B.A. Fellow of Caius College.

[Read May 20, 1839.]

In a former paper I endeavoured to determine in what way a plane
wave would be modified when transmitted from one non-crystallized
medium to another ; founding the investigation on this principle : In
whatever manner the elements of any material system may act upon
each other, if all the internal forces be multiplied by the elements of
their respective directions, the total sum for any assigned portion of the
mass will always be the exact differential of some function. This principle
requires a slight limitation, and when the necessary limitation is intro-
duced, appears to possess very great generality. I shall here endeavour
to apply the same principle to crystallized bodies, and shall likewise
introduce the consideration of the effects of extraneous pressures, which
had been omitted in the former communication. Our problem thus
becomes very complicated, as the function due to the internal forces,
even when there are no extraneous pressures, contains twenty-one
coefficients. But with these pressures we are obliged to introduce six
additional coefficients ; so that without some limitation, it appears quite
hopeless thence to deduce any consequences which could have the least
chance of a physical application. The absolute necessity of introducing
some arbitrary restrictions, and the desire that their number should be as
small as possible, induced me to examine how far our function would be
limited by confining ourselves to the consideration of those media only in
which the directions of the transverse vibrations shall always be accurately
Vol. VII. Part II. Q

122 Mr GREEN, ON THE PROPAGATION OF LIGHT

in the front of the wave. This fundamental principle of FresnePs Theory
gives fourteen relations between the twenty-one constants originally enter-
ing into our function ; and it seems worthy of remark, that when
there are no extraneous pressures, the directions of polarization and the
wave-velocities given by our theory, when thus limited, are identical
with those assigned by Fresnel's general construction for biaxal crystals ;
provided we suppose the actual direction of disturbance in the particles
of the medium is parallel to the plane of polarization, agreeably to the
supposition first advanced by M. Cauchy.

If we admit the existence of extraneous pressures, it will be ne-
cessary, in addition to the single restriction before noticed, to suppose
that for three plane waves parallel to three orthogonal sections of our
medium, and which may be denominated principal sections, the wave-
velocities shall be the same foi» any two of the three waves whose fronts
are parallel to these sections, provided the direction of the corresponding
disturbances are parallel to the line of their intersection. With this
additional supposition, the directions of the actual disturbances by which
any plane wave will propagate itself without subdivision, and the wave-
velocities agree exactly with those given by Fresnel, supposing, with
him, that these directions are 'perpendicular to the plane of polarization.
The last, or Fresnel's hypothesis, was adopted in our former paper. But
as that paper relates merely to the intensities of the waves reflected and
refracted at the surface of separation of two media, and as these inten-
sities may depend upon physical circumstances, the consideration of
which was not introduced into our former investigations, it seems
right, in the present paper, considering the actual situation of the theory
of light, when the partial differential equations on which the determination
of the motion of the luminiferous ether depends are yet to discover, to
state fairly the results of both hypotheses.

It is hoped the analysis employed on the present occasion will be
found sufficiently simple, as a method has here been given of passing
immediately and without calculation from the function due to the internal
forces of our medium to the equation of an ellipsoidal surface, of which
the semi-axes represent in magnitude the reciprocals of the three wave-

IN CRYSTALLIZED MEDIA.

123

velocities, and in direction the directions of the three corresponding dis-
turbances by which a wave can propagate itself in our medium without
subdivision. This surface, which may be properly styled the ellipsoid of
elasticity, must not be confounded with the one whose section by a plane
parallel to the wave's front gives the reciprocals of the wave-velocities,
and the corresponding directions of polarization. The two surfaces have
only this section in common, and a very simple application of our theory
would shew that no force perpendicular to the wave's front is rejected,
as in the ordinary one, but that the force in question is absolutely null.

Let us conceive a system composed of an immense number of par-
ticles mutually acting on each other, and moreover subjected to the
influence of extraneous pressures. Then if x, y, z are the co-ordinates of
any particle of this system in its primitive state, (that of equilibrium
under pressure for example,) the co-ordinates of the same particle at the
end of the time t will become x', y ', z, where x' y z are functions of x y z
and t. If now we consider an element of this medium, of which the
primitive form is that of a rectangular parallelopiped, whose sides are
dx, dy, dz, this element in its new state will assume the form of an
oblique-angled parallelopiped, the lengths of the three edges being
(dx), (dy), (dz), these edges being composed of the same particles which
formed the three edges dx, dy, dz in the primitive state of the element.
Then will

w -{(&"+ (2)' +(£)"}<*-■*'

rl

suppose.

Again, let

o = cos <

(dz')

dx dx' dy dy' dz' dz
dy dz dy dz dy dz

^mW^Wi

dx'\* id 11

T.) + US '

m

Q2

124 Mr GREEN, ON THE PROPAGATION OF LIGHT

dx' dx dy dy dz dz'

(dx') dx dz dx dz dx dz

d#' </#' e?y' </y' dz dz'
(dx') dx dy dx dy dx dy

V {U) + (is) + {dx)\\{dy-) + S|) + Kdy)

or we may write

' _ / _ ftff dx dy dy dz dz
dy dz dy dz dy dz '

., _ dx dx dy dy' dz dz'

/3 = «c/3= -7--7-+-7 L -r- + j T-,

ax dz dx dz dx dz

, j dx' dx' dy' dy dz' dz'
y = aby = -r- -T- + -f- -f- + -j— -v—.
ax dy dx dy dx dy

Suppose now, as in a former paper, that (f> dx dy dz is the function
due to the mutual actions of the particles which compose the element
whose primitive volume = dx dy dz. Since must remain the same,
when the sides (dx 1 ) (dy) (dz) and the cosines a, /3, y of the angles of
the elementary oblique-angled parallelopiped remain unchanged, its most
general form must be

<p = Function (a, b, c, a, (Z, y)

or since a b and c are necessarily positive, also

a = be a, /3' = ac/3, and y = aby,

we may write

<p=f(a\b\c* «',/3', 7 '). (1.)

This expression is the equivalent of the one immediately preceding,
and is here adopted for the sake of introducing greater symmetry into
our formula?.

IN CRYSTALLIZED MEDIA. 125

We will in the first place suppose that is symmetrical with regard
to three planes at right angles to each other, which we shall take as the
co-ordinate planes. The condition of symmetry with respect to the plane
(;/x), will require <f> to remain unchanged, when we change

,> into \ ,

But thus a\ J 5 , c 1 and a evidently remain unaltered; moreover,

,\ become I ,
7 j I-7.

Hence we get

<{>=f(a\ b\ C\ a', (Z'\ 7 '«).

Applying like reasoning to the other co-ordinate planes, we see that the
ultimate result will be

<p=/(a\ b\ c\ «", /r, y s ). (2.)

The foregoing values are perfectly general, whatever the disturbance may
be; but if we consider this disturbance as very small, we may make

X = X + u,

y' - y + %

z' = K + w,

u, v and w being very small functions of x, y, % and t of the first order.
Then by substitution we get

• q^il (—Y ( dv X f_ V, 1

— dx \dx) \dx) \dx) " '

., , rf« (du\ 2 (dv\* idwV ,

bt = l + *Ty + \dy) + Kdy) + (d*) " * + * SU PP OSe -

(3.)

126 Mr GREEN, ON THE PROPAGATION OF LIGHT

, dv dw du du dv dv dw dw
" dz dy dy dz dy dz dy ' dz

1 _ du dw du du dv dv dw dw
~ dz dx dx dz dx dz dx dz

, _ du dv du du dv dv dw dw
' ~ dy dx dx dy dx dy dx dy

we thus see that s„ # 2 , s 3 , a, /3', y, are very small quantities of the
first order, and that the general formula (1) by substituting the pre-
ceding values would take the form

cj> = Function («„ s 2 , s 3 , a, /3', 7')
which may be expanded in a very convergent series of the form

<p = <p + <£, + 3 + <p 3 + &c.

<p (f) { (p z &c. being homogeneous functions of s { , # a , * 3 , a, ft, y of the
degrees 0.1.2.3 &c. each of which is very great compared with the
next following one.

But (p , being constant, if p= the primitive density of the element, the
general formula of Dynamics will give

fffp dx dy dz \~ |« + ~ %v + ~ lw\ = fffdx dy dz (fy, + ty, + &c.)

If there were no extraneous pressures, the supposition that the primitive
state was one of equilibrium would require (p x — 0, as was observed in a
former paper ; but this is not the case if we introduce the consideration of
extraneous pressures. However, as in the first case, the terms (p 3 (p„ &c, will
be insensible, and the preceding formula may be written

fffp d* dy dz j — he + -^ It + -^ Sw J = fffdx dy dz (ty, + %)

IN CRYSTALLIZED MEDIA. 127

Supposing p the primitive density constant, the most general form of 0,
will be

0i = - £ {As, + Bs> + Cs 3 + 2Da + 2E(X + 2Fy'),

ABCDE and F being constant quantities.

In like manner the most general form of 2 will contain twenty-one co-
efficients. But if we first employ the more particular value, (2) we shall get

- 2 0! = As, + Bs 2 + Cs 3
- 2 2 = Gs\ + Hs\ + Is\ + 2P* 2 6' 3 + 2Q*,s 3 + 2Rs l s i

+ La* + M(P+ ivy*.

Or by substituting for s 1} s 2 , s 3 , a, /3', y their values, given by system
(3), continuing to neglect quantities of the third order, we get

- 2 = - 2 0, - 2 2

~ jdu n -r>dv nr . dw

= 2Aj- + 2B- r - + 2C- r -
ax ay a%

+ A
+ B

+

die) + [da;) + \dx) J
UduV (dv\ 2 (dwy\

(4.)

„ (du\* „ (d#\ n jtdwV 9 T>dv dw Q du dw
\dx) \dy) \d%) dy d% dx dx

ndudv , idv dw\ z M fdu dw\ 2 „ (du dvy
dxdy \dz lly) \d* dx) \dy dx)

Having thus the form of the function due to the internal actions of
the particles, we have merely to substitute it in the general formula of
Dynamics, and to effect the integrations by parts, agreeably to the method
of Lagrange. Thus,

128 Mr GREEN, ON THE PROPAGATION OF LIGHT

fffdx dy dx$(j) m

-ffdyd^ASu + A^u + ^Jv + ^w)

\ ax ay a%J \d% ax) \dy ax I J

-ffdxdz{Bh + B{^Su + ( ^v + ^w)

I -.du „dv n dw\ „ T jdv dw\ , ^rfdu dv \ » 1
V M ay as / Vass ay ) \ay ax) )

-ffdxdy{ciw + C^u + fjv + ^Zw)

I r^du n dv r dw\ , T /a"?; «?W\ , -y r (du dw\ t 1

+ ( Q ^ + P ^ + / ^^ + i U + a>)^ + iV te + ^)H

+ ///rf*%rf»l«|G + ^ + (N + B) d ^+ {M + C)g

/\r r>\ ^** /o r\ d'W ]

IN CRYSTALLIZED MEDIA. 129

Neglecting the double integrals which relate to the extreme boun-
daries only of the medium, and which we will suppose situated at an
infinite distance, we get for the general equations of motion,

^df= {G + ^dP +{N + ^ay +{M+C) d¥

(5.)
d 2 v , xr .. d*v . r , „, d 2 v . _ _ d 2 v

? M*- = {N+ A ^dW + {H + B ^df + {L + C) d*

dxdy ' dy d% '

d 2 w .,, j,d 2 w , /T n.d^w , T s^d 2 w

p-d¥-= {M+A) ^ + {L+B) W + {I + C) d^

If now in our indefinitely extended medium we wish to determine
the laws of the propagation of plane waves, we must take, to satisfy
the last equations,

u = of (ax + by + ex + et),

v = fif(ax + by + cz + et),

w = ^/(ttx + by + c% + et) ;

a, b and c being the cosines of the angles which a normal to the wave's
front makes with the co-ordinate axes, a, /3, 7 constant coefficients, and
e the velocity of transmission of a wave perpendicular to its own front,
and taken with a contrary sign.

Substituting these values in the equations (5), and making to
abridge

A'=(G + A)a 2 +(N + B)b*+(M + C)c\

B=(N + A)a 2 + (H+ B)b* + (L + Qc 2 ,

C - (Jf + A) a 2 +(L + B)¥ + (I + C) c*;
Vol. VII. Part II. R

130 Mr GREEN, ON THE PROPAGATION OF LIGHT

B >'= (Z, * F)bc,
E'= (M + Q) ac,
F' = (N + R)ab\

we get

= (A'- e*)a + F'fl + E'y,
= F'a + (JT-/)/9 + ^7,
=.E'a + Z>/3 + (C- e>)y,

(6.)

These last equations will serve to determine three values of e", and three
corresponding ratios of the quantities a, /3, y, and hence we know the
directions of the disturbance by which a plane wave will propagate
itself without subdivision, and also the corresponding velocities of pro-
pagation. From the form of the equations (6), it is well known, that
if we conceive an ellipsoid whose equation is

1 = A' a? + B'f + CV + 2D'yz + %E' x% + ZF'xy* (7.)

and represent its three semi-axes by /•', r", and r", the directions of these
axes will be the required directions of the disturbance, and the corre-
sponding velocities of propagation will be given by

Fresnel supposes those vibrations of the particles of the luminiferous
ether which affect the eye, to be accurately in the front of the wave.

* If we reflect on the connexion of the operations by which we pass from the function
(4) to the equation (7), it will be easy to perceive that the right side of the equation (7) may
always be immediately deduced from that portion of the function which is of the second
degree by changing u, v and w into x, y and z.

«i d d , d . ,

Also, -=- , -=- and -p- into a, b and c.
ax ay dz

This remark will be of use to us afterwards, when we come to consider the most general
form of the function due to the internal actions.

IN CRYSTALLIZED MEDIA. 131

Let us therefore investigate the relation which must exist between
our coefficients, in order to satisfy this condition for two of our three
waves, the remaining one in consequence being necessarily propagated
by normal vibrations.

For this we may remark, that the equation of a plane parallel to the
wave's front is

= ax + by + c%. (a)

If therefore we make

x = x' + a\,

y = y + b\,

% = z + c\,

and substitute these values in the equation (7) of the ellipsoid: re-
storing the values of

A', B, C, D, E', F,

the odd powers of \ ought to disappear in consequence of the equa-
tion (a), whatever may be the position of the wave's front. We thus
get

G = U = / = n suppose,

and P = n — 2L,

Q = /n- 2M,

R = fi - 2N.

In fact, if we substitute these values in the function (4) there
will result

- 20 = - 2^ -20 2 =

2A-j- + 2B-j- + 2C-j-
ax ay dx

+ A

i(du\* (dv\ 2 (dw\ s \

\{dx) + \dx) + Kdlc) j

R 2

132 Mr GREEN, ON THE PROPAGATION OF LIGHT

( (du dv dw\ 2
• + M {{die + Ty + rf^J

T \(dv dw\ 2 dv dw\
\ \dz dy ) dy d%\

__( (du dw\ 2 du dw\
\ \dz dx 1 dx dz J

,-{ /du dv\ 2 du dv\
\ \dy dx) dx dy)'

which, when = A, = B, = C, reduces to the last four lines.

Making the same substitution in the equation (7), we get

1 = n (ax + by ■+ ex'f,
+ (A a 2 + Bb 2 + Cc*) {x 2 + f+ *"), (8.)

+ L(cy - bzf + M(a% - cx) 2 + N{bx - ayf.

Let us in the first place suppose the system free from all extra-
neous pressure.

Then A = 0, B = 0, C = 0,

and the above equation combined with that of a plane parallel to the
wave's front will give

= ax + by + ex (9.)

1 =L(cy - bzf + M(a% - ex) 2 + N(bx - ay) 2 ,

the equations of an infinite number of ellipses which in general do
not belong to the same curve surface. If, however, we cause each ellipsis
to turn 90° in its own plane, the whole system will belong to an
ellipsoid, as may be thus shewn : Let (xyz) be the co-ordinates of any
point j) in its original position, and (x'y'%) the co-ordinates of the point
p which would coincide with p when the ellipse is turned 90° in its

own plane. Then

x 2 + y 2 + z 2 = x' 2 + y' 2 + as' 3 ,

since the distance from the origin O is unaltered ;

= ax' + by' + ess', since the plane is the same;
= xx' + yy + zz, since pOp'= 90°.

IN CRYSTALLIZED MEDIA. 133

The two last equations give

cJ^Tz = ~^~^ = bx-ay = w SU PP° se -

Hence the last of the equations (9) becomes

o? = Lx' 2 + My' 2 + N%\

But

x' 2 + y' 2 + z' 2 = w 2 {(cy - b*y + (az - ex) 2 + (bx - ay) 2 \,
m a, 2 {(b 2 + a 2 ) z 2 + (c 2 + a 2 ) f + (// + v 2 ) of -2 (bcy% + abxy + acx%)\,
= w 2 {(a 2 +b 2 +c 2 )(x 2 +y 2 + « 2 ) - {am + by + c%)* },
- w 2 (« c + y 2 + z 2 ) = x 1 + f + » 2 ,

.: w 2 = 1,
and our equation finally becomes

1 = Lx' 2 + My 2 + Nz' 2 . (io.)

We thus see, that if we conceive a section made in the ellipsoid to
which the equation (10) belongs, by a plane passing through its centre
and parallel to the wave's front, this section, when turned 90 degrees
in its own plane, will coincide with a similar section of the ellipsoid
to which the equation (8) belongs, and which gives the directions of
the disturbance that will cause a plane wave to propagate itself with-
out subdivision, and the velocity of propagation parallel to its own
front. The change of position here made in the elliptical section, is
evidently equivalent to supposing the actual disturbances of the ethereal
particles to be parallel to the plane usually denominated the plane of
polarization.

This hypothesis, at first advanced by M. Cauchy, has since been
adopted by several philosophers; and it seems worthy of remark, that
if we suppose an elastic medium free from all extraneous pressure, we
have merely to suppose it so constituted that two of the wave- dis-
turbances shall be accurately in the wave's front, agreeably to Fresnel's

134 Mr GREEN, ON THE PROPAGATION OF LIGHT

fundamental hypothesis, thence to deduce his general construction for
the propagation of waves in biaxal crystals. In fact, we shall afterwards
prove that the function <p. 2> which in its most general form contains
twenty-one coefficients, is, in consequence of this hypothesis, reduced
to one containing only seven coefficients; and that, from this last form
of our function, we obtain for the directions of the disturbance and ve-
locities of propagation precisely the same values as given by Fresnel's
construction.

The above supposes, that in a state of equilibrium every part of
the medium is quite free from pressure. When this is not the case,
A B and C will no longer vanish in the equation (8). In the first
place, conceive the plane of the wave's front parallel to the plane (yz) ;
then a = 1, b — 0, c = 0, and the equation (8) of our ellipsoid becomes

1 = iux* + A (x 2 + f + z 2 ) + Mz 2 + Ny 2 -,

and that of a section by a plane through its centre parallel to the wave's
front, will be

1 = {A + N)y* + (A + M) z 2 ;

and hence, by what precedes, the velocities of propagation of our two
polarized waves will be

V 'A + iV. The disturbance being parallel to the axis of y.

\/A + M. to the axis of x.

Similarly, if the plane of the wave's front is parallel to the plane
(xz), the wave-velocities are,

\/B + JV. The disturbance being parallel to the axis x.

y/ B -t- L. to the axis z.

Or, if the plane of the wave's front is parallel to (xy), the velo-
cities are,

V C + M. The disturbance being parallel to x.

W+L y.

IN CRYSTALLIZED MEDIA. 135

Fresnel supposes that the wave-velocity depends on the direction
of the disturbance only, and is independent of the position of the
wave's front. Instead of assuming this to be generally true, let us
merely suppose it holds good for these three principal waves. Then
we shall have

N+ A = C + L, M + A = B + L and B + N = C+M;

or, we may write

A-L = B-M=C-N= V . (Suppose.)

Thus our equation (8) becomes, since a 2 + b 2 + c 2 = 1,

1 = M (ax + by + ess) 2 + v (x 2 + y 2 + as 2 )
+ (La> + Mb 2 + iVV) (of + y 2 + s 2 )
+ L (cy — b%) 2 + M (a% - ex) 2 + JV (bx — ayf.

But the two last lines of this formula easily reduce to

(M +N)x 2 + (N+ L)f + (L + M)% 2
+ L \a 2 x* - (by + c%) 2 } + M\b 2 y 2 - {ax + ess) 2 }
+ iV {<?% 2 — (ax + by) 2 },

and hence our last equation becomes

1 = (v + M + N) x 2 + + N + L) f + (v + L + M) «»

+ m (ax + by + ess) 2

+ L \a 2 x 2 - (by + ess) 2 }

(11.)
+ M{b 2 y 2 - (ax + ess) 8 }

+ N{c 2 z 2 - (ax + by) 2 }.

In consequence of the condition which was satisfied in forming the
equation (8), it is evident that two of its semi-axes are in a plane
parallel to the wave's front, and of which the equation is

= ax + by + ess, (12.)

136 Mr GREEN, ON THE PROPAGATION OP LIGHT

the same therefore will be true for the ellipsoid whose equation is (11),
as this is only a particular case of the former. But the section of the
last ellipsoid by the plane (12) is evidently given by

1 = (p + M + N) & + (y + L + JV) f + (p + L + M ) %',

(12, 1.)
= ax + by + c%.

By what precedes, the two axes of this elliptical section will give
the two directions of disturbance which will cause a wave to be pro-
pagated without subdivision, and the velocity of propagation of each
wave will be inversely as the corresponding semi-axis of the section ;
which agrees with Fresnel's construction, supposing, as he has done, the
actual direction of the disturbance of the particles of the ether is
perpendicular to the plane of polarization.

Let us again consider the system as quite free from extraneous
pressure, and take the most general value of 2 containing twenty-one co-
efficients. Then, if to abridge, we make

du _ dv dw _ y

dx = %' a*" 3 * (te~* ;

dv dw du dw _ „ du dv

d% + dy =a ' d% + dx~P' dy + dx~ y '

we shall have

- <h = (f)r+ favv(rt£+ '*tio*x + ■<ft)'er+ '*<w&

+ (« 2 ) a 2 + (/3 2 ) /3 2 + ( 7 2 ) 7 2 + 2 ($y) /3 7 + 2 (ay) ay + 2 (a/3) afi
+ 2(«£)«e + 2(/3£)/3f + S(7f)Y£

+ 2(ari)ar) + 2 (fir,) Pi + 2(yr,)yt,

+ na^ai + i(fip^+i(yt)y^

where (D (a 2 ), &a, are the twenty-one coefficients which enter into 0,.

Suppose now the equation to the front of a wave is

= ax + by + ess.

IN CRYSTALLIZED MEDIA. 137

Then, by what was before observed, the right side of the equation
of the ellipsoid, which gives the directions of disturbance of the three
polarized waves and their respective velocities, will be had from <p i} by
changing u, v and w into x, y and % ;

also -j- , — and -=- into a, b and c.
ax dy d%

We shall thus get

1 = Ax 2 + By* + Czr + 2 Dyz + 2 Ex% + 2 Fxy.

Provided

A = (?) a 2 + (/*) c 2 + ( 7 2 ) ¥ + 2 Q3 7 ) Jc + 2 (f )3) «c + 2 (£7) aJ,

B = (rf) ¥ + (a 2 ) C« + ( 7 2 ) « 2 + 2 (a 7 ) dC + 2 (ijoj &C + 2 (»/ 7 ) a J,
C = (£ 2 ) c 2 + (a 2 ) J 2 + (/3 2 ) a 2 + 2 (a/3) a J + 2 (£a) ic + 2 (£/3) ac,
D= (^)bc + (a 8 ) £c + (j3 7 ) a 2 + (a/3) «c + (ay) ab

+ (ar,)b* + (a£)c? + ((Sr,)ab + (y%) UC,

E = W ac + (/3 s ) ac + (ay) ¥ + (a/3) be + (/3 7 ) ab
+ 03f) « 2 + OD c 8 + (« D «& + ( 7 D f c,

F = (f „) «& + ( 7 2 ) ab + (a/3) c 2 + (a 7 ) ic + (/3 7 ) ac
+ (7?) a 2 + (7,) 6 2 + ( a £) ac + {(ir,) be

But if the directions of two of the disturbances are rigorously in
the front of a wave, a plane parallel to this front passing through the
center of the ellipsoid, and whose equation is

= ax + by + c%,

must contain two of the semi-axes of the ellipsoid; and therefore a
system of chords perpendicular to this plane will be bisected by it;
and hence we get

= (A - C) ac + E (c 2 - a-) + Fbc - Dab,

= (B- C)bc + J)(c 2 - b*) + Fac-Eab.
Vol. VII. Part II. * S

138 Mb GREEN, ON THE PROPAGATION OF LIGHT

Substituting in these the values of A, B, &c, before given, we shall
obtain the fourteen relations following between the coefficients of a ,
viz.

o = (a^, o = oa o = ( 7 a

= («£), = 08ft = ( 7 ,) f
(«?) = - 2 (/3 7 ), Ifa) = - 2 (« 7 ), ( 7 = - 2 (a/3),

(P) = W = (D = 2(« 2 ) + 0?D = a 09-) + (ft) = 2( 7 2 ) + 00l

Hence, we may readily put the function <p 2 under the following
form :

(f ) (? + «! + D 2 + (« 2 ) i« 2 - 4 ^1 + (/3 ? ) (/3 2 - * £D

+ (7 2 ) (7 2 - 4^) + 2(/3 7 ) (07 - 2 «£)

+ 2(« 7 )(« 7 -2/3^) + 2 (a/3) (a/3 - 2 7 0,

or by restoring the values of f, >?, &c, and making G = {$?) L = (a 8 ) &c,
our function will become

„ (du dv dw\ 2 T j (dv dw\~ dv dw \
\dx dy dz) \ \dx dy I dy d%\

__ i (du dw\ 2 .du dw\ ^.[(du dv\ s du dv\
\\dz dx) dx dz) \\dy dx) dx dy)

pf tdu dw\ (du dv\ du (dv dw \ 1

\\dz dx) \dy dx) dx \zd dy)\

(12.)

„J/c?u dw\ (du dv \ dv (dti dw\\

\\dz dy) \dy dx) dy \d% dxlj

■pUdv dw \ (du dw \ dw (du d v \\

\\d» dy) \dz dx) dz \dy dx))''

+

and hence we get for the equation of the corresponding ellipsoid,

1 = G {ax + by + c&y + L(bz — cyf

+ M(a% - cxf + N(ay - bxf + 2P(cx - az)(ay - bx)

(13.)
+ 2 Q (bz — cy) (ay — bx) + 2 B (bz — cy) (ex - az).

IN CRYSTALLIZED MEDIA. 139

But if in equation (8) and corresponding function (A), we suppose
A = 0, B = and C = 0, and then refer the equation to axes taken
arbitrarily in space, we shall thus introduce three new coefficients, and
evidently obtain a result equivalent to equation (13) and function (12).
We therefore see that the single supposition of the wave-disturbance,
being always accurately in the wave's front, leads to a result equivalent
to that given by the former process ; and we are thus assured that by
employing the simpler method we do not, in the case in question,
eventually lessen the generality of our result, but merely, in effect,
select the three rectangular axes, which may be called the axes of elas-
ticity of the medium for our co-ordinate axes. From the general form
of 0, it is clear that the same observation applies to it, and therefore the
consequences before deduced possess all the requisite generality.

The same conclusions may be obtained, whether we introduce the
consideration of extraneous pressures or not, by direct calculation. In
fact, when these pressures vanish, and we conceive a section of the ellipsoid
whose equation is (13) made by a plane parallel to the wave's front, to
turn 90 degrees in its own plane, the same reasoning by which equation
(10) was before found, immediately gives, in the present case,

1 = Lx' 2 + My' 2 + Nz' 2 + 2 Py'z' + 2 Qx'z + 2 Ra'y' (14.)

for the equation of the surface in which all the elliptical sections in their
new situations, and corresponding to every position of the wave's front,
will be found.

Lastly, when we introduce the consideration of extraneous pressures,
it is clear, from what precedes, that we shall merely have to add to the
function on the right side of the equation (13) the quantity

(Aa 2 + Bb 2 + Cc 2 + 2Dbc + 2Eac + 2 Fab) (x 2 + f + z 2 ),
which would arise from changing u, v and w into x, y and as. Also -r- , -r- ,

-j- into a, b, c, in that part of (p x which is of the second degree in u, v, w,
agreeably to the remark in a foregoing note. Afterwards, when we

S2

140 Mr GREEN, ON THE PROPAGATION OF LIGHT, &c.

determine the values of A, B, &c., by the same condition which enabled
us to deduce the system (12, 1), we shall have, in the place of this
system, the following:

1 = K(a* + if + O - \La? + My 2 + iW + 2 Py% + 2 Qxx + 2 Rxy\. .

(15.)
= ax + by + ex,

which is applicable to the more general case just considered.

VIII. On a Portion of the Tertiary Formations of Switzerland. By
D. T. Ansted, Esq. M.A., Fellow of Jesus College. Fellow of
the Society and of the Geological Society ; Professor of Geology
in King's College, London.

[Read May 20, 1839.]

The Tertiary formations of Switzerland are singularly deficient in
most of those points which have rendered the contemporaneous deposits in
other countries of Europe so attractive and important. The beds, for the
most part, vary Jbut little in mineral structure : they seem to have been
accumulated rapidly, and under circumstances little favourable to the
preservation of organic remains, and the few fossils that are known to
occur, possess none of that definite character, which elsewhere indicates
with sufficient clearness to what well-known group the one in question
was anterior, and what beds were anterior to it. Owing, perhaps, to this
want of determinate character, and partly, also, no doubt, to the superior
interest of the strangely contorted secondary beds, which form the
principal mass of the great mountain district always within sight, it
has happened that travellers in general have neglected to examine care-
fully the great valley of Switzerland, and I am not aware of any detailed
account in our own language of so considerable a portion of European
Tertiary Geology.

I am not able, indeed, myself to add much to the small amount of
our knowledge on this subject, but anxious at all events to direct atten-
tion to it, I have ventured to lay before the Society a few observations
made during a stay of several weeks at Lausanne, in the summer of 1838.
In order to do this most effectually, I shall first consider the nature of

142 PROFESSOR ANSTED, ON THE

the Tertiary beds occurring in what is called the Great Helvetic Basin,
and occupying the space between the High Alps and the Jura chain.
I shall afterwards proceed to remark upon the various smaller basins met
with in the Jura district itself, and partially rilling up the valleys be-
tween the different ranges of that mountain chain.

From whichever side Switzerland is entered, whether from France,
Germany, or Italy, no traveller, not even the most indifferent about
geological phenomena, can have failed to notice the physical structure of
the country, or the effect to the eye of that series of deposits concerning
which I am about to speak. The high range of mountains to the South,
nearly terminated at each end by the two highest of the European
mountains, Mont Blanc, and Monte Rosa — the continuation of these
lofty eminences toward the North-East, forming the " High Alps," and
extending into the Northern Cantons of Switzerland — the less lofty but
still considerable elevations running parallel to this principal range in
the West of Switzerland towards France, and known as the "Jura"
chain — all these very remarkable and strikingly beautiful mountain chains
surround a tract of land comparatively level and rich in every thing
that can administer to the wants or luxuries of man ; and it is this
cultivated district, this comparative plain in a land of mountains,
which marks out the extent of the Swiss Tertiary deposits, and has
hitherto been, as I observed, almost neglected by the geologist. It
requires, perhaps, to have been on the spot to understand the tempta-
tion offered by the near proximity of such mountains; but those who
have been there, and have hurried on with all the enthusiasm and ex-
citement of novelty to breathe the pure and exhilirating mountain air,
will wonder but little that the plains have been neglected, and that the
Tertiary Geology has given place to the Alpine.

It was hardly an effort of philosophy which induced me to labour in
the less trodden field : — a conviction that I could not hope to make much
way where so many and far superior and more practised geologists had
preceded me, may indeed have induced me the more readily to be contented
in a less distinguished sphere, but my expeditions from Lausanne were
necessarily short, and my opportunities limited.

TERTIARY FORMATIONS OF SWITZERLAND. 143

Thus circumstanced, my observations will be found to relate chiefly to
the South-Western part of the Helvetic Basin, and not at all to the more
interesting portion extending Northwards and Eastwards from Berne,
and already somewhat minutely described in a work, published in 1825,
by Professor Studer, of Berne. My excursions were, as I have said, con-
fined to a small part of the Canton of Freyburg, and the greater part
of the Pays de Vaud. The limit of this district to the South is the
lake of Geneva. The Eastern and Western boundaries are sufficiently
defined by the abrupt elevation of mountains, forming the flanks of the
High Alps on the one side, and of the Jura on the other.

The high road from Freyburg to Vevey is nowhere at any great dis-
tance from the line which separates the tertiary beds from those secondary
ones upon which they lie uncomformably, but the actual junction at any
point I did not perceive, as the country is for the most part covered
up, and the geological phenomena obliterated. Close to Vevey, however,
in a valley cut by a small stream coming down to the lake, we obtain
a glimpse of the extreme tertiary beds to the East, and it will be perhaps
best if, commencing with these, we trace the collocation of the beds as
they are exposed on the North side of the lake of Geneva, and mav
be observed in travelling from Vevey towards Lausanne and Geneva,
westwards.

Close to the town of Vevey there occurs a hard conglomerate, very
coarse where it rests on the older rock below, but becoming gradually
finer, until after a few miles it is replaced by a very fine sandstone, which
spreads over the whole centre of the valley of Switzerland, and is the great
tertiary deposit of which I have chiefly to speak. Of these beds, the coarser
conglomerate is known generally by its German designation, " Nagelfluhe,"
while the nature and peculiarities of the finer sandstone (which is the
most widely spread and extensive of all the European Tertiaries) are in-
dicated in the name " Molasse," by which the soft, incoherent tertiary
sandstone of this country and Germany is designated.

The thickness of the Nagelfluhe is various, but never very great. From
near Vevey it may be traced towards the West for about a couple of
miles, gradually becoming a finer deposit, and imperceptibly changing
into the Molasse, without any definite line of separation.

144 PROFESSOR ANSTED, ON THE

It would be extremely difficult to lay down the limits of this bed
with accuracy, although the thickness cannot be any where very great ;
I could not discover a single spot where the dip could be taken, but as
the whole seems to have undergone a change of position by disturbances
connected, doubtless, with the uplifting of the mountain chain, no single
observation of this kind, even if it were made, could possess much value
in the way of determining the mass of the deposit.

On the other hand, the thickness of the Molasse, although equally
difficult to determine, must be enormous, and if calculated in the ordi-
nary way, allowing for its being repeated once or twice by faults, will still
appear almost incredible. Extending across the valley of Switzerland for
nearly five and twenty miles, and inclined often at angles varying from
15 to more than 50 degrees, rising sometimes into hills four thousand feet
above the sea-level and more than two thousand above the general level
of the country, we cannot escape the conclusion that it is a mass of
vast thickness, even after making every allowance for the effects of dis-
turbance.

I am inclined, however, to think that much of this appearance of enor-
mous thickness is owing to the deposit having been formed on a consider-
able slope, and not on a horizontal or nearly horizontal plain, and that thus
its almost uniform inclination is not owing entirely to disturbances of
the substratum, but also to the circumstances of deposition. If we imagine
the formation to have been commenced when the level of the valley
of Switzerland was below that of the sea, and that sandbanks rapidly
formed on a shelving coast at some distance from the shore, were gradually
raised by successive small elevations, and afterwards when the general level
of the land was above that of the sea, that the elevations had gone on
from time to time till the present state of things was produced, we
should have very similar phenomena presented to view, viz. an enormous
mass of sandstone, appearing to possess a dip that would multiply its real
thickness tenfold, and ranges of hills at some distance from the former
coast-line.

The main difference between the Nagelfluhe and the Molasse, consists
in the mechanical difference between a coarse conglomerate and a fine sand-
stone, but interstratified with the Molasse there occur here and there beds

TERTIARY FORMATIONS OF SWITZERLAND. 145

of lignite, which add much to the geological interest and something to the
economical advantages of the district under consideration. There is also
found in the West of the Canton of Vaud, not far from the lake of Neu-
chatel, a white building-stone containing much calcareous matter in its
composition, but circumstances prevented me from paying that attention
to so interesting a stratum which it well deserves from the geologist. It
will be found forming a hill close to the little town of Thierrens, and I
observed it in one spot dipping about 40 degrees to the South- West*.

Having thus described the mineral composition of the different strata
observed, I come now to speak of the general outline of the country, and
the deductions to be drawn from considering the physical features produced
probably by disturbances acting after the beds had been deposited.

Although, in comparison with the stupendous chain of the Alps, the
central and more cultivated portion of Switzerland is properly designated
as a valley, yet even in this valley there occur eminences which in a
more level country might well be called mountains. About five miles
from Vevey, and to the west of the coarse conglomerate called Nagelfluhe,
there rises a hill of Molasse to the height of nearly 4000 feet, and a chain
of hills may be observed extending from this (which is called the Tour de
Gourze) towards the North-East, whose heights are successively, 3000,
4000, and 3500 feet above the sea. In speaking of these altitudes, how-
ever, it must not be forgotten, that the level of the lakes of Geneva and
Neuchatel is considerably more than twelve hundred feet above the sea, and
thus the hills do not in reality form such striking features in the landscape
as others of no greater actual elevation, but rising from a lower plateau,
in other countries, and under different circumstances. Imbedded in the
sandstone of which these hills are composed, there occurs in the line of
the hills, and about ten miles North of Vevey, one of the beds of lignite
already alluded to, and we are enabled accordingly to determine the dip
with some accuracy, at all events in this spot ; I observed that it was very
considerable, certainly more than 50°, and its direction variable, though on
the whole Easterly, being here, and in one or two other places along the
line, towards the South-East, in a few others North-East, and some-
times nearly due East.

* This would appear to be a local deviation from the general dip of the district.
Vol. VII. Part II. T

146 PROFESSOR ANSTED, ON THE

If leaving the chain of hills just alluded to we advance along the
banks of the lake of Geneva, towards the West, we come to a parallel but
less elevated chain, beginning about ten miles from Vevey, near the town
of Lausanne, forming a ridge of sand-hills whose summits are about 2500
feet above the level of the sea, and the ridge continues at nearly the same
elevation for a distance of at least 15 or 20 miles to N. E. Here, as
before, the dip is towards the South-East, and generally as much as 45°.

Between Vevey and Lausanne another bed of lignite of some thick-
ness is worked. The bed is exposed in consequence of a mountain torrent
having cut its way through the Molasse, close to the spot where the
lignite crops out to the surface. It is thus worked in chambers, from the
right bank of the stream to the outcrop, which is at no great distance.

If we return now, and continue our course along the banks of the
lake still further to the West, we shall find a third time indications of
a similar North and South range, commencing at a celebrated point de vue,
called the Signal of Bougi, from which may be enjoyed one of the most
beautiful and picturesque prospects in this part of Switzerland. The
chain of hills commencing here, is continued at an elevation of little less
than 3000 feet for many miles, parallel to the mountains of the Jura.

In several places the dip of the Molasse may be observed in the neigh-
bourhood of this, as of the other parallel lines of elevation, and is generally
South-East. The opportunities however of obtaining dips are so very
rare, and except where the lignite occurs, the bedding so obscure, that
if it were not for the uniformity wherever the inclination can be clearly
made out, I could hardly venture to lay much stress on a series of observa-
tions, so few in comparison with the large extent of country over which
they are spread.

On the whole, however, we seem to have in this Southern portion of
the Molasse of Switzerland, three distinct and tolerably well-marked lines
of elevation, all parallel to the mountain chain of the Jura, from which
also they all dip. The upheaving of this latter chain (the Jura) subsequent
to the formation of the High Alps, seems to have been the means by which
the peculiar physical features of these tertiary beds were in a great
measure produced. Doubtless there have been great changes effected
by the action of the elements upon beds so soft, and often almost inco-

TERTIARY FORMATIONS OF SWITZERLAND. 147

herent; but still the great amount of dip considered in connection with the
parallel ranges alluded to, gives us sufficient reason for referring to elevations
as the original causes of the more remarkable phenomena.

The Tertiary Geology of Switzerland is but little assisted by the
consideration of those organic remains which are peculiar to, or discovered
in the various beds. The Molasse is so exceedingly barren of fossils,
that during many weeks which I spent in the immediate neighbourhood
of great natural sections of it, I did not on any occasion find a single
specimen indicating organic structure. The Northern beds are, however,
rather more prolific, and offer sufficient evidence that this vast mass of
sand was accumulated under sea-water. There is a list of fossils in the work
by Professor Studer, already alluded to, which includes the following
marine genera, — Mactra, Cytherea, Cardium, Pecten, Trochus, Cassis,
Terebra, Buccinum, and Conus. These were most of them found in various
parts of the Cantons of Zurich and Lucerne.

Although, however, the general character of the bed, as well as
the discovery of such a series of fossils, would induce us to place the
whole formation among marine deposits, yet with regard to the bands of
lignite, the evidence is so entirely the other way, and points so clearly
to a fresh-water origin, that I think the only way of reconciling the
apparent anomaly is to suppose the former existence of considerable
streams rushing down from the mountains, and bringing with them vast
quantities of vegetable, intermixed with some animal remains, which
might be deposited at the mouth of a river in consequence of a bar,
or extensive sandbank.

The shells found in the lignite, and embedded in the sandstone
immediately adjacent, are chiefly Helix, Planorbis, Lymncea, and Unio ;
but the specimens are so much broken, that the exact species can hardly
be determined. Besides these, I was fortunate enough, on one occasion,
to discover a portion of the sternum of a chelonian reptile, probably a
turtle, although such fossils are, I believe, extremely rare, and I did not
hear of any other remains of Reptiles during my stay in the South of
Switzerland. The lignite is generally hard with a clean conchoidal fracture
and brilliant lustre, and is a good deal used for fuel. It is met with in

T 2

148 PROFESSOR ANSTED, ON THE

beds several feet in thickness, but not extending far in any direction ;
and these beds alternate usually with thin marls, which are often quite
white in consequence of the enormous abundance of crushed shells
belonging to land and fresh-water species, which often completely hide the
marl, and cover the surface of the lignite.

SECTION I. ACROSS THE GREAT VALLEY OF SWITZERLAND.

Level oj Lake of Gtnna.

Level of the Sea. scale of dUlancet i of an inch i

.... 7 , . i to a mile,

of heights .} inch >

From the annexed section some idea may be formed of the relative
positions and magnitude of the three lines of elevation already alluded to
as existing in the Molasse, and the place which the two beds of lignite
occupy : the dips are principally towards the East, and more or less
with a Southerly tendency, but the amount varies, and is, I think,
generally most considerable in the neighbourhood of the High Alps,
towards which the strata incline.

I have now only to add a few words more on this part of my subject ;
viz. to point out, so far as 1 am able, what remains to be done for the
more complete illustration of the Tertiary Geology of Switzerland.

In the first place, there occurs a question of great interest, and one
which requires, probably, very accurate research to determine: viz.
whether the lines of elevation to which I have directed attention were
really caused by upheaving forces, or merely by denudation — whether,
in a word, there are lines of fault, or anticlinal axes, corresponding to
the lines of elevation. In the next place, it would be extremely interesting
to identify, if possible, the two beds of lignite — a task which I was
unable to perform ; and lastly, it is possible that, barren as the sandstone
is of fossils, some may yet be discovered, by which we may declare
with certainty the actual geological age of the formation. With regard
to this latter point, I shall have a few words to add at the conclusion of

TERTIARY FORMATIONS OF SWITZERLAND.

149

this paper, and must for any further information refer to the work already
alluded to, published at Berne by M. Studer.

Quitting the wide expanse of the great Helvetic Basin, I wish next
to direct attention to the circumstances connected with the tertiary
valleys of the Jura, and more particularly to the valley of la Chaux de
Fonds, which may serve indeed as a type of the rest.

The villages of la Chaux de Fonds and le Locle, at the two extremities
of the same valley, are the richest, the most populous, and, in some respects,
the most remarkable of any in Switzerland. They are situated near
the frontier of France, one in the Northern and the other in the Southern
part of a valley which is about ten miles long and one broad, extending
in a North-Easterly direction, at an elevation of more than two thousand
feet above the level of the sea. There is no outlet to the valley for drainage
at either extremity, and its general appearance, as well as geological
structure, show clearly that it was formerly the bed of a mountain lake,
resembling in all probability those still existing in the Jura, such as the
Lac de Joux, the Lac de St. Point, and one or two others. As I first
visited la Chaux de Fonds from Neuchatel, and afterwards entering
the valley at its South- Western extremity, passed le Locle and again
reached the village on my journey Northwards, I will first describe
in a few words the section across the Jura, and then the peculiarities
which present themselves in tracing the beds in the direction of the
valley's length.

SECTION II.

Level of the Sea,

ACROSS THE PRINCIPAL RANGES OF THE JURA.

Ttte it Rang:

City i
VaUengy. ^^^ of ft

Scale i inch to a mile.

City and hake
NeucMtel.

Immediately on leaving the town of Neuchatel the road begins to
rise, although the passage across the first range of the Jura is rendered more
easy by its following the course of a transverse valley, which brings
a mountain torrent from the first and most Easterly valley to the lake
of Neuchatel. The naked walls of rock exhibited on each side of the
road show clear marks of the violent dislocation which must have accompa-
nied the upheaving of the mountain chain, and we can trace easily the

150

PROFESSOR ANSTED, ON THE

direction of the anticlinal axis, and the contortions of the strata near the
highest part of the range. The descent on the Western side is rapid
but not very long, and brings us quickly to the little town of Vallengy,
which is built upon a considerable bed of gravel, the superstratum of
a valley, without doubt, of tertiary formation. The valley thus covered
up is a fair specimen of many of those occurring between the two most
Easterly parallel ranges of the Jura : they are for the most part desolate
and barren, now and then watered by a small stream, but then only present-
ing a little pleasing scenery close to the water's edge. I should imagine
that they had been formed rather by the action of submarine currents
depositing gravel, than by any regular subsidence of transported matter
in a lake.

Crossing this valley, whose breadth is here nearly two miles, we come
again to the secondary rocks of the Jura, and the road passes over the middle
and highest range of those mountains. At a very high elevation, and en-
closed between two ridges of nearly 4500 feet, occurs a second small valley,
much more narrow and insignificant than the one before mentioned ; and
after having crossed it, there is a sudden and rapid descent, leading down to
the third and principal valley, that of la Chaux de Fonds, the examination
of which was the main object of my excursion. In physical features, as well
as geological structure, this valley has all the character of a lacustrine
deposit, left dry, either by the silting up of a mountain lake, or gradual
evaporation for want of a sufficient supply of water. There is certainly no
outlet for water, and scarcely a single running stream in its whole extent.

The village of la Chaux de Fonds is near the northern extremity of the
valley, and about midway between the mountains on the east and west. It
is built partly upon a small bed of clay and marl, marked (a) in Section 8,

SECTION III. ACROSS THE VALLEY OF LA CHAUX DE FONDS.

Jura.

Juea.

If!'

Level of the Lake of NeucMlel

Scale 2 inchei to a mile.

TERTIARY FORMATIONS OF SWITZERLAND. 151

and partly upon a fresh- water limestone, the upper beds of which alternate
with the marls above. On each side of this band of limestone, marked (b),
there comes out another series of marls (c), resting upon the Molasse (d)
which is here of no great thickness, and overlies a portion (probably the
lower part) of the chalk formation {e), immediately below which in this
part of the district are the upper oolite beds of the Jura (y).

In the uppermost of all these beds, resting on the fresh-water limestone,
there have been discovered, in digging foundations for houses, several frag-
ments of bones, among which were teeth in tolerable preservation. These
bones, being examined by competent anatomists*, have been referred to the
following genera : — Anoplotherium, Palceotherium, and Lophiodon, Hippo-
potamus, Camelopardalis, Equus, Deinotherium, Elephas, and Rhinoceros.
To the bed containing these fossils, and the circumstances under which they
occur, I am desirous now of directing attention.

The bed I have already sufficiently described as a black earthy deposit,
alternating with calcareous bands. It is pretty regularly stratified, and I was
struck with the probability there seemed of its having been formed while
the lake, which doubtless once covered the whole valley, was so far dried up
as to resemble a marshy pond, in which the bones would be preserved as in
a peat bog. Of the species determined, I believe five have been identified
as occurring also in the Paris Basin ; the others would seem to belong to a
more recent period, and perhaps we should rather refer to the tertiary beds
of Bordeaux, and the valleys of the Garonne and Loire, than to the neigh-
bourhood of Paris for analogies. The Miocene period of Mr Lyell has
already been suggested by that gentleman as the probable date of the Jura
tertiaries, and the discovery of these fossils would tend to confirm his
opinion.

The Molasse, however, being the substratum, and resting immediately
upon the cretaceous beds, it is clearly an older deposit, perhaps existing
as the bottom of an ancient sea, before the disturbances and elevations,
which formed the valleys of the Jura, and raised them to their present
position, took place.

* Most of the specimens were determined by Professor Agassiz, and many of them sent to
Paris to be compared with the fossils examined and named by Cuvier, and found in the Lower
Tertiary formation of the Calcaire grossiere.

152 PROF. ANSTED, ON TERTIARY FORMATIONS OF SWITZERLAND.

In conclusion, the Tertiary Geology of the South-West of Switzerland
may be said to be separable under three heads ; first, the great deposit of
Molasse, which appears, from all we can tell, to be of marine origin; secondly,
the fresh-water marls and lignite bands occurring in the Molasse, but very
local, and apparently near the upper part ; and thirdly, the overlying beds of
marl and limestone in the valleys of the Jura, which alone can be compared
with the better developed systems in other parts of Europe; but since, from
the general dip of the sandstone, that portion of it in the Jura valleys would
seem to have been the earliest formed, there is no reason why the overlying
beds there should be very much newer than the lignite near the Alps. The
period therefore to which the Molasse must be referred, still remains in
doubt. It also results from the dips and observations recorded, that the so
called great Helvetic Basin is in fact no basin at all, but a vast accumula-
tion of sandstone, formed probably upon an inclined plane, and then tilted
to a greater or less angle into its present position. The smaller valleys are
indeed true basins, but the structure of many of them, especially the most
Easterly, is a point, I think, yet to be determined.

It is obvious that much remains to be done in determining the true
geological relations of the Molasse, its fossils, and the varieties of its dip ;
and I would especially direct attention to the limestone near the Southern
extremity of the lake of Neuchatel, which is the most promising of
any part of this tiresome formation. Should there be found here any
fossils, they must possess great interest; and I regretted extremely the
want of opportunity which prevented me from examining accurately
the whole neighbourhood. It is very accessible, being close to the high road
between Yverdon and Moudon, and certainly deserves the attention of any
geologist travelling in that part of Switzerland.

Should anything I have said lead to the determination of this and other
points in Swiss Tertiary Geology, my object in bringing this paper before
the Society will be fully accomplished.

D. T. ANSTED.

Jesus College,
March 1841.

IX. On the Quantity of Light intercepted by a Grating placed before a
Lens; and on the Effect produced by Interference. By the Rev.
Philip Kelland, M.A., F.R.SS.L. & E. late Fellow of Queens'
College, Cambridge ; Professor of Mathematics in the University of
Edinburgh.

[Read March 30, 1840.]

From the remarkable appearances presented by the interception of a
part of the light proceeding from a small luminous body towards the object-
glass of a telescope, it may very naturally be supposed that vibrations are
suffered to exist, which would otherwise be destroyed by interference ; and
that consequently a less quantity of light is stopped by the grating than
that which is actually incident on it. That light actually appears from the
application of the grating, where there would be little or none without it,
is most certain ; and that this circumstance arises from the want of inter-
ference, alluded to above, there can be no doubt. Should we then expect,
notwithstanding the cause to which we attribute the phenomenon, (not to
speak of the phenomenon itself) to find exactly the same quantity of light
on the other side the lens, or at least in its field of view, as would corre-
spond to the spaces left open by the grating ?

When first asked my opinion on this subject, I had no hesitation in pro-
nouncing that, previous to calculation, I should expect to find more light
transmitted through a grating, than in proportion to the space left un-
covered. My idea of the matter was this : certain vibrations are not
destroyed when the grating is applied, which would be destroyed in the
contrary case ; whereas there is nothing to affect those spaces from which
vibrations are excluded, so as to render this nugatory. This reasoning,
subsequent consideration convinces me is incorrect. It appears that, although
the stoppage of vibrations by the wires does bring into operation that

Vol. VII. Part II. U

154 PROFESSOR KELLAND, ON THE

which must otherwise have been destroyed, yet the same stoppage causes to
disappear certain of the vibrations corresponding with the uncovered part,
which would, in the contrary case, appear in the aggregate of all the
motion.

My attention was called to this subject by Professor Forbes, who
has been prosecuting an experimental enquiry into the effect produced
by screens on the transmission of radiant heat. The curious fact which
he has established relative to the difference in amount of the stoppage
produced in light and dark heat, — or at least in two different kinds of heat,
which he has found to be operated on very differently in other matters —
promises to give us an insight into the characteristic properties of light
and heat, provided it appear that one kind of heat is, in the case before
us, acted on in the same manner as light is, in like circumstances. But
perhaps it is too much to hope that we shall distinguish betwixt light
and heat, uncertain as we are of the intensity of the former, by which
its nature might be contrasted with that of the latter. It may then be
expected, rather, that we shall be put in the way of distinguishing between
heat and undulations ; distinction being, if I mistake not, absolutely
necessary, as well as obviously pointed at, by the very experiments which
seem most strongly to identify the two with each other.

I forbear, however, entering on this subject at present, although I
am deeply interested in it, as well on account of its intrinsic importance, as
of its bearing on my own views of the Theory of Heat. I shall therefore,
without further preface, proceed to the question in hand.

Our Problem is this : —

A series of equal parallelograms are placed before a lens, to find the
whole quantity of light received on a screen, placed perpendicular to the
axis of the lens at its focus.

The solution of the Problem for rinding the intensity at any one in-
dividual point will be found in Airy's Tracts, p. 328, at the foot of
Art. 83.

The expression is this :

'!/ 2 (

p (e + g) 7T
sin t — - — r- 5 — m <

2nqf Xb J Wpe \b I \ T p (<? + g) n

sm Xb

AGGREGATE EFFECT OF INTERFERENCE. 155

The notation is as follows :

e is the breadth of one of the openings between the wires ;

g the breadth of a wire ;

p, q the co-ordinates of the point, measured along the screen from the
focus of the lens,

p being perpendicular to the wires ;

m the number of openings and of wires.

To obtain the whole quantity of light, then, we must multiply this
expression by 4:dpdq, and integrate between the limits and oo .

Let the result of the integration for q give

■He* r»i /sin aV/ 01 " V e)

sin 1 1 + - 1 mx\

w r*p (nr) . / V\ ) b y writin s * for w

\ sin 1 + - ) x I

Then dp = — dx, so that the expression becomes

ire

He Cdx (*™-Z)' ( ^rmx y f _ g
J \ x J \ sin rx I e

1. Now first, we must integrate this expression on the hypothesis that
the aperture is uninterrupted, or that m — 1. I shall make use of the well
known formula of Laplace : viz.

/

» a cos qx .dx _ir _ aq

a? + x* 2

the particular value of a being 0.

Thus r^-f

Jo x i 2a

cos xdx

' X?

-co

cos 2 xdx

X s

&c.

u2

- ~e- a

2a

£■

2a

2a "

&c.

156 PROFESSOR KELLAND, ON THE

Supplying these values, we obtain,

1 r" v 1 — cos 2x

f M 5 * 5 ) -if* 3

a?

7T

= — (1 — e' ia ), a being always =

7T

2 ;

. •. the intensity on this supposition is He — , or as we will write it, to

denote that e stands now for the semi-aperture, HE

TV

We find then, that the whole quantity of light incident is exactly
that which corresponds to the open space between the bars, no effect
being produced by interference, or its destruction.

2. Next, as a particular case, in order to make the process intelligible,
we will find the illumination when the breadths of the wires are equal
to the openings between them.

If, for instance, there be two wires, the general formula gives

J \ x ) Vsin %xi

„ r» fsmx\ 8 . , „ ,
= He (— — I 4cos 2 2#a#

= 4iHef — (sin 8 a; — 4 sin 4 a; + 4 sin 6 a?)

= 4 He f -, (- — ;rCOS 2x + tcos 4>x - -cos 6x )
Jo x* \4 8 4 -8 /

-*ra.(*-i+§)

= it He.

AGGREGATE EFFECT OF INTERFERENCE. 157

But e = - E ; since the original aperture is divided into four equal parts,
two of which are appropriated to the openings, and two to the wires.

.-. The intensity = ^ HE

= - the result found for the intensity when there are
no wires.

3. Thirdly, let us retain the hypothesis, that the breadths of the
wires are the same as the openings between them, but suppose the number
of wires and of openings to be any number whatever, (m).

Write — , for sin x:

2v/- 1

then the expression (sin xf r 1 ? |~ j is put under the form,

1 (0 fl-«Y ( ** - *"" V

" 1 \ e + e- 1 ) '

or

Let (^ — ^-J = oofl 4 — •+ «!0 4m - 4 + &c.

+ a4 m - 2 0- (4m - rt ;

.-. 4 " 1 - 2 + 0" 4m = O 4 "' + a 1 , "-* + 0,0*—*+ ... + a4 m -20- (4m_4)

+ 20 o 4 "" 2 + 2« 1 4 '"- 4 + ... + 2a 4m . 2 0- u '"- 2)
+ OO0 4 — 4 + ... + a tm -z0- im ;
from which we obtain by equating coefficients,

1 = 00,

= 0,+ 20 o>

= 2 + 20*1 + 0o,
= 03 + 20 2 + 0„

158 PROFESSOR KELLAND, ON THE

— 2 = tti m + 2ftg m -i + (tin _2,

2(l lm -;; + W 4m _3,

#4m-2 = !•

These results give

a Q m 1, «! = - 2, «2 = 3, «3 = — 4, &c.

asm., = — 2m, «2 m = — 2 + 4m — 2m + 1 = 2m — 1,

<W, = - (2m - 2) &a « 4m _ 8 = - 2, a 4 ™-a = 1-

By substitution, therefore, the general expression

tt r°= t /sin #\ 2 /sin rmx\ % ,

iie / d# . — becomes

Jo V x ) \ sm rx 1

— 2 {cos (4m — 2) x — 2 cos (4m - 4) x

+ 3 cos (4m — 6) x — &c + (2m — 1) cos 2x - m\

or _ I **± i c -««-»« _ 2 e -(««-«« + se-w»-«« _ & c . + (2m - 1) e"*" - nl
4 « l

But 1 - 2 + 3 - &c. + (2m - 1) - m = ;

•. the expression gives ; observing that a = 0,

+ r-H>(4m- 2 - 2 4m- 4 + 3 4m - 6 - &c. + 2m - 1.2).
4

Now 4m - 2 - 2 (4m - 4) + 3 (4m - 6) - &c. + (2m - 1) . 2
m 4m {1 - 2 + 3 - &c. + (2m - 1)}
- 2 {I s - 2 2 + 3 2 - &c. + (2m - l) 2 }
= 4m 2 — 2m (2m — 1) = 2m.

AGGREGATE EFFECT OF INTERFERENCE. 159

Hence the expression becomes

- Hem:
2

that is, - H x space left uncovered.

Or, which is the same thing, since m . 2e = E,
the expression is - . - HE,
or, one half the quantity of light which would fall were there no grating.

4. To give one more particular case, we will take that in which
g = 2e, or the breadth of the wire is double that of the opening between the
wires.

Our expression then becomes He f ( ) (— = ) dx,

Jo \ x I \ sin 3x J

and - 4 (sin *)■ ( *? *"* )' = (0 - <H)' fl ~ £") ' = ( g ~ '"" X

y J V sm 3x I y } \ e 3 - o- 3 I ye* + i + e-J

Assume this expression to be expanded in the form

a,^"- 4 + « 2 6m - 6 +

+ a 2 0- (6m - 6) + «,0- <8m - 4) :

the terms from the beginning and end of the series having obviously equal
coefficients,

. 0S. _ 2 + e - 6m = (04 + 29 2 + 3 + 20- 2 + 6-*)

x \a l 6 6 "'- i + ch& m -« + + a 2 0- (6m - 6) + a 1 0- (6ra - 4, }

= a,& m + a 2 &'"-* + OvO 6 "-* + « 4 6m - 6 + a^'"-^...

+ 2a l e 6 "'- 2 + 2a#'"- 4 + 2a 3 & m - e + 2a t 6 6m - b +...

+ Sflifi 8 " -4 + 3(h& m -' i +- 3a 3 d 6m - s +...

+ 2a 1 6 6m - 6 + 2ck& m - 8 +...

160 PROFESSOR KELLAND, ON THE

Equating coefficients, we obtain
«i = 1,
«2 + 2«, =
th + 2« 2 + 3«i =
« 4 + 2as + 3«2 + 2a, =
a 5 + 2« 4 + 3«s + 2«2 + a, =
a 6 + 2« 5 + 3a 4 + 2a 3 + a 2 =

«*,+, + 2a 3m + 3a3 m _, + 2(h m _ 2 + flfc._» = - 2,

= 0;

,*. a x = 1, a 2 = — _2
«3 = 1, a 4 = 2, a 5 = — 4
a 6 = 2, a 7 = 3, a 8 = — 6

a, = 3, a, = 4, a u = - 8

Suppose this law to hold true for any three consecutive terms ;

that is, let <h n = »»

chn + 1 = n + 1,

«3 B + 2 = - 2«- 2;J

• '. «3»+3 = 4w + 4 — Sn — 3 - 2w + 2«
= ft + 1,
a3„+4 = — 2n — 2 + 6n + 6 — 2w — 2 — »

= n + 2,

03„ +5 = — 2» — 4-3» — 8 + 4» + 4-«-l
= — 2« - 4 :

hence the law holds good for the next greater consecutive terms, and is
consequently general.

AGGREGATE EFFECT OF INTERFERENCE. 161

Also the middle term is

(hm-i = 2m.

We thus obtain as the value of the intensity,
He

2

*He

4a

J dx — {cos (6m - 4) x - 2 cos (6m - 6)x + ...+ m\

{ c -<«»-«>«_ 2£? - (6 "'- 6)a + ... + m]

rrHe

j {1 - 2 + 1 + 2 - 4 + ... to (3m - 2) terms + m\

4 a
ir He

{1 . (6m - 4) - 2 (6m - 6) + ... to (3m - 2) terms}.

Now 1 — 2 + l+...+ m is obviously half the above expansion, when
= 1, and is consequently zero.

Also l.(6m- 4) - 2 (6m - 6)+

= 1 (6m - 4) + 2(6m - 10) + + m(6m - 6m - 2)

- 2 (6m- 6)- 4 (6m - 12)- -2(m - 1) (6m - 6m - 6)

+ 1 (6m - 8) + 2 (6m - 14) + + (m - 1) (6m - 6m- 4)

= 1 .(12m - 12) +2 (12m - 24)+...+ (m - 1) (12m - 12m - 12)

- 2 (6m- 6) — 4 (6m -12)- - 2 (m -1) (6m - 6m - 6)

+ m (6m — 6m — 2)
= 2 m.

Hence the whole intensity is - Hem = - H x space left uncovered, the
same result as before.

5. Lastly, let us take the most general case, of a grating in which
the thickness of the bars bears any proportion whatever to the spaces left
uncovered.

Vol. VII. Part II. X

162 PROFESSOR KELLAND, ON THE

Adopting the general expression, we have now to find the value of
fsin rmx\ 2 (0 - fl" 1 ) 2 (&* ~ •"

, . so (smrmx\ 2 (0 - 0-') 2 [9* - 0- r "Y

( sin *) llhTTFJ or i— U^H •

Assume

/firm __ *J-"n\ 2

fL. — ^_j = 0iK— 1) + ^©trc— »> + + p-MtV",

.-. 2 ™- 2 + 0-2™ = ^™ + « 2 2 '' ( '"- l) + a30 2r( '"- 2) + ...

+ 2r(m - 2) + ...
which by equating coefficients, gives

ch - 2 = 0,

fl 3 — 2« 2 + 1 = 0,

a 4 — 2a 3 + « 2 = 0,

#m + 1 — 2 Cf m + fl! m _ 1 = — 2,

=0:

hence a 2 — 2, 03= 3, a 4 = 4, &c.

and a m+ i = 2»a - (m — 1) — 2
= »a — 1

that is, the coefficients form an arithmetic series, increasing up to m,
and then diminishing down to 1.

By multiplying by the factor (0 — 9~ l f, we obtain

02r(m-l)+2 1 2Q 2r(m-2)+2 X.,.4 /ftg 2 + . . . 4. 0-2r(m- l)+2

+ 0- { *^ ui) + ZQ-V'"^** + ...+ mQ- 2 + ...+ d*'^-" ~ 2

= 2 {cos (2/-/W - 1 + 2)# + 2cos(2rw - 2 + 2)#+...
+ m cos 2a; + (m — 1) cos (2r — 2) x +...+ cos [2r(?« - 1) - 2]x\
— 4 {cos 2r(m — l)x + 2 cos 2r(m - 2)x +...+ (m ~ 1) cos 2rx]

- 2m = K.

AGGREGATE EFFECT OF INTERFERENCE. 163

Hence the intensity is

He r*dx R= _ Z**Zi e -^ i+«. + *-»=i+!>. «.... + me -*«
4 J & *«

+ (m - 1) e-<- Sr - 2)a + (m - 2)<r (4r - 2)s + ... + e -^^-*"

- 2 [<?-*<"-»* + g e -*<»-»)<. + ... + ( m _ i)^-^™]

— m\

7v He

4a

{1 + 2 + + m + (m - 1) + ... + 1

- 2(1 + 2 + ... + m - 1) - m}

7T

i?e

+ —7— .{l.(2r.m- I + 2) + 2(2rm - 2 + 2) + 3{2rm-3 + 2)

+ ... + (*» - 1) (2r -f- 2) + m . 2
+ (m - \)(2r - 2) + (m - 2) (4/*- 2)+ ... + 2r(m - 1) - 2

- 2 \_2r .m — l + 2.2r(m—2) + ... + (m - 1) 2r~]\

Ti e

— — — . \4>r(m — 1) + 2 . 4r (m — 2)+ ... + (m — 1) 4r

+ 2m — 4r (m — 1) - 2 . 4r («» — 2) — ... - (m — 1) 4r|
■n-Hem ttII

2 2

x space left uncovered.

Thus it appears that the whole quantity of light is not at all affected by
the diminution of interference. For we obtain, whole quantity of light on
the screen : that which falls on the object-glass :: area of the uncovered part
of the glass : whole area of the glass.

It is unnecessary to dwell on this result. That it is a strong confirmation
of the undulatory theory, as far as regards two hypotheses respecting the
intensity, and the vibrations in different directions, cannot be doubted.

x 2

164 PROFESSOR KELLAND, ON THE

The common assumption, that the intensity is measured by the square
of the excursion of a vibrating particle, although bearing a great air of
probability, is still not so obvious as to derive no benefit from a confir-
mation such as our conclusions tend to give it.

The hypothesis respecting the intensity of vibrations in different di-
rections, and at different distances, as stated by Mr Airy, is this: that
a vibrating particle transmits vibrations equally in all directions, but with
an intensity varying inversely as the distance. This hypothesis is not alto-
gether conformable to our conclusion, which appears to require that vibra-
tions transmit forces equally in all directions, and to all distances. Fortu-
nately, none of the approximate results deduced from either hypothesis
are vitiated by it, since the variation of distance is not taken into consi-
deration in the solution. Of course, these observations are based on the
supposition, that the division of a complete wave into elementary portions,
in the manner always employed to effect the exhibition of results dedu-
cible from a change of circumstances in the mode of transmission, is
allowable. My object, at present, being rather the demonstration of a pro-
perty of undulations, than an application to the theory either of light or
heat, I have contented myself with alluding to the bearings of the result
to which we have arrived. What has been said will be confirmed by the
following problem, with which the preceding is intimately connected.

" The whole intensity of light reflected at the surfaces of two plane
mirrors, inclined to each other at any angle, is not altered by the interference
of the light from the one mirror with that reflected from the other."

■»■

To this problem we shall annex the same limitations, and apply the
same processes as to that already solved. That is to say, we shall conceive a
lens placed before the mirrors, so as to bring the reflected light to two foci
lying in a line perpendicular to that which bisects the angle between the
mirrors.

Let C be the projection of the line of intersection of the mirrors ; O, P,
the foci to which the rays from the mirrors respectively converge. Then
each wave on leaving the lens will be a portion of a sphere, of which the
centre is the point of convergence.

AGGREGATE EFFECT OF INTERFERENCE. 165

Let b be the radius of the sphere = BO, AO =f, AM = p, AE = x,

EB = y + f, BE being perpendicular to AC. Then the vibration at the
point M due to a vibration C at B is

c sin ~ (vt t BM).

A

But #M* = (p-f- y y + x 2

= (p-JJ- s(p-/)y + « , + y 2

.-. I?il!f = 1? - £ ~^ . y nearly.

Hence the vibration at M produced by the upper mirror, as far as its
projection on the plane of the paper is concerned, is

2cfy sin — {vt - (p -/*) - ¥ + 2(p -/) - y\.

Also, if B be not in the plane of the paper, BM 2 becomes

(p -/- yf + ** + **> or (p -fj -2(p-f)y + b\
as before.

166 PROFESSOR KELLAND, ON THE

Hence, the expression for the vibration is

cf dy sin — - (vt — B + * '' y J

In the same way, the vibration due to the second mirror is

- /*» , sin '- » i4ff ^ sin »T (,< _ g - a + /> </ "A

ir(p+f) X b X V 2b )

B" being equal to Z? + 33 :

so that the whole vibration at the point M is

Msin^-Z? ^-^-^ )
XV 2b J

where M and N are the coefficients of vibration due to each mirror
respectively.

If we expand these in terms of the common circular arc, which is

AGGREGATE EFFECT OF INTERFERENCE. 167

we obtain, calling this arc 9 for the sake of brevity,

Imcos = fJ£pD + jvcos = *»<* -/) + 4p/l s . n

+ {i^sin^^y^-iv-sin^^-^+^cosg;

so that the intensity at the point is represented by

M 2 + A ra + 2 MN cos — Pte+f) §

X o

We must next integrate this expression between the limits — oo and
+ oo for p.

now/:^^/^^^^; ^-/)^^) .^-/)

- ** >. I 1 " C ° S X * ft J (j, -/)'

= — — — 1 1 — e~ \ b I , a being equal to zero,
c*X 2 ft 2 «* g+f

Similarly, f° N*dp = c 2 Xft(# +/).

•/-co

Hence we find that the whole effect of each mirror is proportional to its
aperture : which result is strongly confirmatory of the general character
of our calculations.

Lastly, MNcob^P^D
=V (i , 2 _ /2 sin x ft sm x ft cos x ft '

ft* 2tt

g>-(f -r> \ ^ — ; — °"t r-/™""x

168 PROFESSOR KELLAND, ON THE

Now the circular functions in this expression are

/> x IT 27T ff + f

(COS aj — COS ap) COS ap, Calling — j~ > «•

But by Laplace's Formula,

and the integral between our limits is merely the double of this ;
.•. the integral of the term

ilfiV cos op = A {cos afe' a ^~ l - i - £e~ 2aA/rT !

= A {cos af cos af— <J — 1 sin af — ± — ^ (cos 2af- •/ — 1 sin 2a/*) I

= A {COS 2 a/* - i — 1 COS 2af— <s/ — 1 (sin a/cOS a/ — ^ sill 2a,/)}

=
a very remarkable result.

If M be not in the plane COP, there is a factor — in M and JV,

which amounts to the factor f I — j dq or -g- in the final result.

/^sin <7A\ 2 7 ^^

* 1 ri n r\w

q

We find then that the whole intensity is the sum of the intensities
due to each of the mirrors separately. Should the form of the function
M* as integrated for the whole space be objected to, the only reply is,
that one or other of two things must be supposed ; either 1° that
the integration for spaces perpendicular to the plane of the paper would
take away X, or 2° that the intensity is a function of the length of the
wave. In either case our conclusion is correct. There is evidently some
factor required to render this result of the same dimensions as that with
which we set out. Perhaps I am not warranted in assuming, from the
coincidence of my results with the principle of vis viva, and their con-
sequent probability, that this factor is not variable from point to point.
When the question first arose in my mind respecting this matter, I thought
to answer it at once by an appeal to the transformations effected by the
" Differential Calculus to any indices." Although the result of this appeal
is very far from satisfactory, I do not think it will be deemed an un-
pardonable digression to take it here.

AGGREGATE EFFECT OF INTERFERENCE. 169

The principle assumed as the basis of adulation is this*: " The effect of
any wave in disturbing any given point, may be found by taking the front
of the wave at any given time, dividing it into an indefinite number of
small parts, considering the agitation of each of these small parts as the
cause of a small wave, which will disturb the given point, and finding, by
summation or integration, the aggregate of all the disturbances of the
given point, produced by the small waves coming from all points of the
great wave."

I took, then, the simplest case which can be conceived, viz. that of an
infinite plane wave. There can be no doubt that the result in this case
ought to be the following : that the disturbance produced is the same in
intensity as that corresponding to one of the points in the disturbing
wave.

Let b be the perpendicular distance of the given point from the wave,

then it is evident that if c sin — (vt — x) be the disturbance of any point in

the wave, the effect produced, according to the above principle, will be re-
presented by

2tt J crdr sin — (vt — y/r 2 + b 2 ) x some quantity.
Nor is it less evident that the result actually is c sin — (.vt — b).

A

What therefore is the multiplier in question? If it is not constant, it
must be some function of r.

Denote r 2 by %, and let the multiplier bey (ss) :
then our equation assumes the form

7T . f* d% sin — - (vt — \A + U)f (as) = sin — (vt — b).
But by the very elegant theorem of M. Liouvillef

/o" <f> (x + a) a*- 1 da= (- l)^/^ (x) dx^,

* Airy's Tracts p. 267- Traite de la Lumiere, par C. H. D.Z. (Huygens), p. 17. A. Leide,
1690. t Journal de l'Ecole Polytechnique, 21« Cahier, p. 8.

Vol. VII. Part II. Y

170 PROFESSOR KELLAND, ON THE

we obtain, if we write 2 A^~ l for,/ (as), a for U ;

f &- 1 d% sin — (vt - y/% + a) = ( - Vf&l <p («) dor

= (— i)>/ sin x ( vt ~ v/ ") rfa ^'
and consequently, from the equation above,

1A{— iy*£ /" sin-^(i^ - y/a)daf l = sin -^ («£ - Va).

•'A A

This equation is satisfied (apparently) by making n = 0, A = - ,

and consequently f(%) = — , /(f) = — 2 .

The case is, however, of too doubtful a character to warrant us in
adopting the conclusion. One thing alone I infer from it, that if any power
of the distance (not of r) be assumed as the factor, it must be the inverse
square. It would require that we should retrace our steps, and investigate
the different formulas corresponding to this hypothesis, before Ave could
speak positively on the subject. I have only to add to this discussion on the
probable coefficient of vibration, that an approximation has been made use
of in the value of the distance between the disturbing and disturbed points,
as it appears within the circular function. The approximation amounts in
fact to supposing the wave elliptical, instead of circular. In the second
problem I find that the square of this distance, being substituted within the
circular function for the distance itself, leads to precisely the conclusions we
have obtained. It is possible, therefore, that the omission of our factor, and
the approximation made use of within the circular function, exactly coun-
terbalance each other.

I cannot conclude without repeating my conviction of the importance
of results such as those which Professor Forbes has just announced. It
appears that the effect of scratching a piece of rock salt, &c. is to alter its
power of transmitting heat in such a manner, that heat of a low tem-
perature, or dark heat, is transmitted in greater proportions than before. If

AGGREGATE EFFECT OF INTERFERENCE. 171

then the two kinds of heat correspond, the one to vibrations, or transmission
due to vibrations ; the other to transmission due to excess of elasticity, our
analysis teaches us to expect that the quantity of the former kind stopped
by the wires or scratches should be in exact proportion to the space covered
by them, whilst we should hardly expect to find any considerable stoppage
effected on the latter. Thus I am led to hope that the Theory which I pro-
posed in the Transactions of the Cambridge Philosophical Society, Vol. vi.,
pp. 274, and seq. and subsequently developed in my little work on the
subject, will be strengthened in some points, although I am far from
expecting that it will be confirmed in all. Perhaps subsequent results may
render it necessary to modify our hypotheses, but at present I do not know
that experiment is very far in advance of theory. I cannot conclude with-
out expressing my conviction that the masterly researches of Professor
Forbes will have the effect of setting right several errors even in the Theory
of Light, which have crept in from the difficulty of subjecting that branch
of philosophy to strict measurement.

P. KELLAND.

Edinburgh,
Jan. 23, 1840.

Y2

X. On the Foundation of Algebra. By Augustus De Morgan,
F.R.A.S. F.C.P.S.; of Trinity College; Professor of Mathematics
in University College, London.

[Read Dec. 9, 1839.]

The extent to which explanation of the meaning of the symbolical
results of Algebra has been carried within the last half century ; the com-
plete interpretation of all which formerly appeared incongruous; the sepa-
ration, as it was called, of the symbols of operation and quantity, which
amounts to the use of an algebra in which the symbols represent something
more than simple magnitude ; — will for some time to come suggest inquiry
into the logic of this many-handled instrument of reasoning, which seems
to be capable of presenting, under fixed laws of operation, all the results
which arise from very distinct primary conceptions as to the things
operated upon.

When several different hypotheses lead to results which admit of a com-
mon mode of expression, we are naturally led to look for something which
the hypotheses have in common, and upon which the sameness of the method
of expression depends. A comparison of the properties of the ellipse and
hyperbola would bewilder the imagination, under any of the distinct defi-
nitions which might be given of the two curves ; nor would the mind
rest satisfied until it had discovered the reason of the similarity which exists
between these properties.

Algebra now consists of two parts, the technical, and the logical. Tech-
nical algebra is the art of using symbols under regulations which, when this
part of the subject is considered independently of the other, are prescribed
as the definitions of the symbols. Logical algebra is the science which
investigates the method of giving meaning to the primary symbols, and of
interpreting all subsequent symbolic results. It is desirable that the word de-
finition should not enter in two distinct senses, and I should propose to retain

174 PROFESSOR DE MORGAN, ON THE

it as used in the art of algebra, applying the terms explanation and interpreta-
tion to denote the preparatory and terminal processes of the science. Thus a
symbol is defined when such rules are laid down for its use as will enable us
to accept or reject any proposed transformation of it, or by means of it. A
simple symbol is explained when such a meaning is given to it as will enable
us to accept or reject the application of its definition, as a consequence of
that meaning: and a compound symbol is interpreted, when, having
occurred as a result of explained elements, used under prescribed defini-
tions, a necessary meaning can be given to it ; the necessity arising from the
tacit supposition that the compound symbol, considered as a new simple
one, must still be subject to the prescribed definitions, when it subsequently
comes in contact with other symbols. The last words may need the remark,
that though we sometimes appear to interpret a symbol merely for the
purpose of explaining a result, ye we know that such interpretation would
be subsequently rejected, if the use of the symbol, under the prescribed
definitions, were not found to be logically admissible.

A symbol is not the representation of an external object absolutely, but
of a state of the mind in regard to that object ; of a conception formed, for
the formation of which the mind knows that it is or was indebted to the
presence, bodily or ideal, of the object. Those who do not remember this,
the real use of a symbol, are apt to dogmatize* declaring one or another
explanation of a symbol, that is, the signification by it of one or another
impression produced on their own minds, to be real, true, natural, or neces-
sary : it being neither one nor the other, except with reference to the par-
ticular mind in question. To take a very simple case, and one which bears
upon our subject, let us imagine that we form successively a conception of
the absence of all definite magnitude, followed by one of the existence of a
certain magnitude, say a line of given length. The mind of one person may
pass from the one to the other by imagining the given length to be instanta-
neously generated, no one portion of it coming into the thoughts before
or after another ; that of a second may make the transition by imagining a
point to move from one extremity to the other : while that of a third may
dwell rather on the relative position of the two extremities, and may think

* Of course, I use this word in its primitive sense, without any censure implied : the very-
sentence in which the word occurs is, and is meant to be, dogmatical.

FOUNDATION OF ALGEBRA. 175

more of B attained by motion from A, than of the quantity of length in
AB. All three would use, perhaps, the same modes of expression : and I
suspect* that there could be detected, among persons who think about first
principles, a very considerable degree of variety in the points of view under
which fundamental words suggest their objects ; while as much exists, but
could not as easily be found, among those who have studied the exact
sciences, without paying particular attention to their foundations.

A symbol may thus denote either magnitude, operation, by which mag-
nitude is attained, or the conception of one extreme arrived at, the other
having been the previous object of contemplation. The earlier f algebraists
most certainly dwelt on the first notion ; a + b is with them the result of
an operation, in which the method of obtaining it is so completely for-
gotten, that the result a + b is actually obtained by a distinct operation.

It seems to me that Sir William Hamilton, in his very original and
methodical memoir on algebra as the science of pure time, has adopted a
view of the third kind. I cannot see why the whole paper might not be as
easily applied to succession of points in a line, as to succession of epochs in
time. Succession, that is to say continuous succession, might be made the
fundamental conception in both cases ; and if such were the author's inten-
tion in the use of the word time, I should be very glad to maintain after him
that one of the explanations which suffice to convert technical into logical
algebra, has been fully established in his memoir. But, if any thing more
physical\ be intended by the distinguished author, and if some of his
phrases are to be interpreted as of his asserting algebra to be the science of

* In a short biographical account (which I have before me, in a private communication) of
the late Mile Sophie Germain, whose papers on the theory of elastic surfaces are well known,
it is asserted that she could never form the conception of space, except by the means of time :
this was her own mode of expressing, to the writer of the notice, a state of mind by which
he accounts for another fact, namely, that she had very little aptitude for pure geometry, and
a great attachment to the theory of numbers.

t See my Calculus of Functions, sect. 245.

{ This word is here improperly used ; but I refer to the notion of those who would have
made geometry a part of mixed mathematics : that is, if the algebra of Sir W. Hamilton would,
in the opinion of those just alluded to, also have been a part of their mixed mathematics, and
if Sir W. Hamilton should admit that they have as much reason, his terms being understood in
his own sense, for their location of his algebra as for that of geometry, I should then say that
the word used in the text is allowable.

176 PROFESSOR DE MORGAN, ON THE

pure time, I should then cite him as an instance of the dogmatism already
alluded to : and the more readily, that by the association of the word with
his labours, I may claim to have purified it, for the purposes of this paper,
from the dyslogistic associations usually connected with it.

The modern algebraists usually dwell on the second notion, namely that
of operation ; and this I shall adopt in the present paper, not only as the
most common mode of conception, but also as being equally capable of con-
nexion with either of the other two. Imagine the process, whatever it may
be, by which we pass from the contemplation of to that of a ; then if a
represent a line, we can consider, as a result of our process, either the posi-
tion of one extremity with respect to the other, or the quantity of length
intercepted between the two.

I separate the following maxims from the rest as being equally ap-
plicable to the symbolical algebra which we have, and to any other
which we might have. For it must never be forgotten that, though our
present inquiry includes only the possible explanations of one given tech-
nical algebra, the subject may and probably must end in the investigation of
others, or at least in the extension of the present one.

1. A simple symbol is the representative of one process, and of one
only.

2. All processes, how many soever, may be looked at in their united
effect as one process, and may be represented by one symbol.

3. Every process by which we can pass from one object of con-
templation to another, involves a second by which we can reinstate the
first object in its position : or every direct process has another which is
its inverse. To complete the separation of these maxims from all others,
I propose some considerations connected with the possible extensions of
technical algebra.

The system of explanations which proceeds on the supposition that
length affected by direction is the primary object of contemplation in
algebra, is well known as to its history by Professor Peacock's Report
to the British Association, and as to its present state by the Treatise on

FOUNDATION OF ALGEBRA. 177

Algebra of the same author*. But in this branch of logical algebra
the lines must be all in one plane, or at least affected by only one modifi-
cation of direction : the branch which shall apply to a line drawn in any
direction from a point, or modified by two distinct directions, is yet to be
found.

It is obvious that our power of making the preceding application
of algebra is co-ordinate with that of assigning a symbol Q, such that

a + bQ — «, + b^Q gives a — a x and b = b u

An extension to geometry of three dimensions is not practicable until
we can assign two symbols, Q and w, such that

a + bQ + cm — a, + b x Q + c x w gives a — « 1} b = b x and c = c, :

and no definite symbol of ordinary algebra will fulfil this condition.
Again, in passing from a; to — x by two operations, we make use in
ordinary algebra of one particular solution of

<p 2 x = — x, namely cpx = \/ - 1 . x.

An extension to three dimensions would require a solution of the equation
<p*x = — x, containing an arbitrary constant, and leading to a function of
triple value, totally unknown at present.

A general solution of cp^x = ax can be expressed when any particular
solution -&X is known. For if f-arf-^x be the general solution, we have

cpx =f , sr !! f- ] x =faf~ l x — ax, or fax = afx:

so that it is only necessary that f and a should be convertible. Since then
( - l)^x is a particular solution of (jy'x = — x, a general solution is

f\ - 1 \f~ x x} where f(-x) = -fx. But with our very limited knowledge
of the laws of inversion, no solution which we can now express in finite
terms will afford any help. Our means of expression must be augmented
before we can hope to overcome this difficulty : or, as in most other cases

* Professor Peacock is the first, I believe, who distinctly set forth the difference between
what I have called the technical and the logical branches of algebra. The second term, I am
aware, is a very bad one, and I should be glad to see a better one proposed ; but I prefer
technical to symbolical, because the latter word does not distinguish the use of symbols from
the explanation of symbols.

Vol. VII. Part II. Z

178 PROFESSOR DE MORGAN, ON THE

of the kind, our difficulties recur in a circle; the means which we have
used to propound a possible method require the problem itself to be solved
before they can be successfully used.

Let the object of contemplation be simple magnitude of any one kind,
as in the arithmetic of concrete quantity. The process which must precede
all others is what we call selecting one magnitude for consideration.
Previously to this step, we have no object under our perceptions, and
may write as the representative of this preceding state, and as the
recognition of its existence. This first magnitude we may call 1, and
the operation of transition from one state to the other we may denote by

+ 1. The contemplation of simple existence, and of the possibility of ex-
pressing it by a spoken symbol, suggested the earliest definition of unity —

M0NA2 ecrTi, icaO' r/v o eKaarov run bvrwv ev Xeyexai. If we represent

our present state by (0 +1), we may consider that with respect to any other
possible magnitude our position is what it was when we denoted it by 0.
If we now denote it by 0', we may, as before, make the transition from
0' to 0' + 1, which implies that we have further taken into consideration a
new magnitude of the same amount.

This result, (0 + 1) + 1, we may, if we please to consider it as
attained by one operation, signify by + 2 : and so on. Using the symbol
— to denote the process by which we retrace our steps, we have all
that is necessary to express addition and subtraction. The principle which

1 wish here to enforce is, that addition is connected with the symbol in
a manner which requires us to imagine that we start from one magnitude,
as it were from a new 0, and renew* the process by which we passed from
the first to that magnitude.

Let us now suppose that modified magnitude is under contemplation,
and let the simple symbol a denote a line measured in a given direction
from the zero point 0. In this zero of space, which admits of an infinite
number of positions, we seize more clearly than before that notion which,
as to simple magnitude, is not easily admitted as necessary, and may
seem rather fanciful : namely, that every magnitude attained may, as

* Any one who doubts the justness of this fundamental position should add six to four on
his fingers, having previously refreshed his notions of six and four by the same process.

FOUNDATION OF ALGEBRA. 179

to future addition, be considered as a new zero. We are now to assume
that,

1. Parallelism and sameness of direction are meant to be identical
terms ; that is to say, the two directions conceivable on any one of two
parallels are severally the same as the two directions on the other.

2. Every simple symbol represents a line given in length and direction :
thus a = b means that the lines a and b, equal in length, have also the
same direction. And the process implied in + a is the transference
of a point from the position to a given length in a given direction.

We can now find the necessary meaning of (0 + a) + b ; necessary,
on the supposition that the technical algebra is
to become logical on the explanation of the sym-
bols before us. Let 0A and 02? represent the
lines symbolized by a and b : if then we take
A, at which we arrive by the process + a, as
a new zero, and proceed with it in the same
manner as in performing + b on the old zero,

we draw AC parallel and equal to 02?, whence 0C being symbolized
by c, we have with reference to the first zero,

+ c = (0 + a) + b = (0 + b) + a.

I need not further dwell on the connection of addition and subtraction
in arithmetic with the processes called by the same names in this ex-
planation. I shall only here suggest that perhaps the words direct zero
process and inverse aero process might occasionally be found useful *.

The usual method of defining the process of addition by reference
to the diagonal of a parallelogram is convenient, but destructive of all
true analogy. The fundamental theorem of statics suffers from the same
method of statement.

I now proceed to the process of multiplication, which will readily
be seen to be connected with unity in precisely the same manner as is
addition with zero. If b be formed from unity by the train of processes
+ 1 + 1 + 1, we consider a as a new unit, and let the symbol ba represent

* In my Calculus of Functions (sect. 12, 13, 17) will be found some analogies connecting
simple addition with zero, and multiplication with unity.

ISO

PROFESSOR DE MORGAN, ON THE

the same operation on this new unit, or + a + a + a. Similarly, if by
the line 1 we mean a line having the length and
direction 1, and OA and OS by a and b, and if we
take Oias a new unit, and perform on it the opera-
tions by which we pass from 01 to 02?, that is, take
an angle AoC equal to 10 J?, and let 0C be in
length the result of the arithmetical operation on Oi
and OB, — then 0C must be represented by ab. The
processes of multiplication and division might be called
the direct and inverse unit processes.

There is now nothing particular to be said about the four operations,
or the simple powers, with positive or negative, whole or fractional, real
exponents, or any combinations of them. The interpretation of a + b\/- 1
follows in the usual manner.

In illustration of the propriety of considering symbols as functions of
zero or unity for purposes of addition or multiplication, it may be advanced
that unless we do so, we change the meaning of the terms direct and in-
verse as we proceed from the lower to the higher parts of the science. Un-
questionably, if ever we have a right to assume a clear conception of this
distinction, it is in the comparison of addition with subtraction, and of mul-
tiplication with division ; but for all that, a + x and a - x are not inverse
functions, considered with respect to x, though they are so with respect to a.
And similarly of ax and a -=- x. When we come to the symbol x", then, and
then only, do we begin to describe inversion correctly : for we usually
consider this as a function of x and not of n, when we assert Xn to be the
inverse. But if we considered this as a function of n, the inverse would be
log n : log x.

The separation, as it is called, of the symbols of operation and quantity,
is a method of explaining technical algebra as simple in its character as the
preceding. Let the fundamental object of conception be (p (x — nh), n
being infinite, which stands in the place hitherto occupied by 0. Let*
2 (j> (x + ah) represent the train of operations by which we pass from
(\>(x — oo h) to cf> (x + a — lh), or

* In the common method of treating this subject, the inverse symbol is made to precede
the direct one. Several adaptations of notation are necessary before we can exactly represent
the common methods.

FOUNDATION OF ALGEBRA. 181

<f>(x - oo h) + + (p(x - h) + <px + (p(x + h) + + <j>(x + a — \h).

The inverse operation, or rather the operation by which <p (x + ah) is
obtained from 2<£(# + ah), is either 2 {<p (x + a + \h) - <p (x + ah)\,
or 2 <p (x + a + 1 h) — 2 cj> {x + ah), and may be symbolized either by
A2 <p (x + ah) or 2A <p (x + ah).

The proper way, however, of considering this class of extensions may
not be as a simple explanation of technical algebra, (though it might be
regarded in that point of view,) but as an extension of technical algebra
itself, in which new explanations of the direct and inverse unit process are
used co-ordinately with the one already established. If we agree to signify
by v°, v 1 , v 2 , &c. a new progression of operations, in which the zero and its
processes remain subject to the usual definitions, nothing prevents us from
supposing that the prescribed definitions of the unit process may remain
true if v° be made the unit, v 2 being derived from v ' by the same train of
operations as v 1 from v°, and so on. Neither is it impossible that the same
laws of convertibility and distribution may exist between compound opera-
tions, in which different units are employed, as are laid down in the pre-
scribed definitions relatively to the different unit processes suggested by
simple magnitudes.

Let v° = <px, and

V 1 = a a (px + «i (p {x + h) + a 2 (p (x + 2h) +

where a , a x , &c. may be functions of h, but not of x. Technical algebra may
be carried to its full length under these explanations, and many deve-
lopements may be and have been simplified by their means. It is not my
intention here to write a treatise on this subject : my object is, to point out
that the logic of each and all of these explanations is the same ; no mode of
arriving at any one explanation differing from that of any other in the funda-
mental, and what ice may call the arithmetical, part of the subject. It is certain
that the discovery of inverse operations is not yet complete : this must be
reserved until such time as the branches, which adopt length modified by
direction as the explanation of simple symbols, are properly connected with
that technical algebra, in which various unit processes are used co-ordinately
with the same zero process.

182 PROFESSOR DE MORGAN, ON THE

It may perhaps be worthy of note that the series

do + a x x + a % x l +

may be considered as v e" when v = in the equation

V€" = a u e v x « l6 u+log * + a s e v+2logx +

I now return to the purely algebraical question. It is in our power to
avoid all ambiguity in results, by simply prescribing that every symbol
shall express not merely the length and direction of a line, but also, the
quantity of revolution by which a line, setting out from the unit line, is
supposed to attain that direction. When this is done, I shall use a double
sign of equality to denote it. Thus, if we denote by (a, 9) a line of a length
a, which has made the revolution 9, it is allowable to write

(a, 9) = (a, 9 + 2tt) = (a, 9 + 4tt),

but not

(a,9) = = (a,0 + 27r) = = (a,0 + 4tt)

As long as we neglect this additional prescription, great care will be
requisite to prevent our falling into error. While exponents transform
lengths into lengths, and directions into directions, no great caution is re-
quisite : but when, as we shall presently see, an exponential process causes
the exponent of a length to affect that of direction, or vice versa, the follow-
ing fallacy of a continental analyst, mentioned by Professor Peacock in his
Report, is frequently likely to occur. Stripped of unnecessary details, it is
as follows :

or e ~ iir3n ' = 1, an absurd result.

The answer is very simple : if no extension of explanations be contem-
plated, i*W^i is not necessarily = 1, since it may have an infinite number
of values. If the extensions be made, and if = merely denote sameness of
direction, the same thing is true ; for l 2 W=i or (e 2 ™^) 2 ™^" 1 has an in-
finite number of values, of which one only ( € <>)2™V=i is = 1 : and the same
fallacy might be thus propounded ;

^/z* = + x, y/x* = — x, therefore x = — x.

FOUNDATION OF ALGEBRA. 183

But if = imply sameness of revolution, it is not true that e 2 *^- 1 = 1,
except in length.

The interpretation of A^~ x might be easily attained from prescribed
definitions, and from their necessary result

e ev=i — C os 9 + sin 9 \/ - 1 ;

nor would this step be logically objectionable. It would, however, be more
satisfactory if something like an a priori interpretation, or simple explana-
tion, could be given. I do not consider the following as complete, but it is,
as far as it goes, of a new character.

Conformably to definitions, we must have

{(log a, 0)V=I}^ . {log a, 6}-' = (- log a, - 9),

where by (log «, 9) is meant a line of the length a, and amount of revolu-
tion 9. Now we cannot suppose that the first operation changes the sign of
log a only, and the second that of 9 only : for this would be to make the
operation ( y~ l mean different things in different places. We must pro-
pose some operation of permanent form, which being twice performed will
make the alteration required.

From the definitions, it follows that

(log a, 0) x (0, 9) = (log a, 9),

whence (log a, 9) must be the product of two functions, one of a and
the other of 9, the first of which is known, being e loga or a, and the second
of which must be of the form E e , since by definition

(0, 9) x (0, 9') = (0, 9 + ff).

Hence aJE e , or a(0, l) e , is the representative of a line a, inclined at an
angle 9. If then we make cos 9 and sin 9 mean nothing more than the pro-
jecting factors of a length inclined at the angle 9 upon the axis of the unit
line and its perpendicular, we have

(cos 1 + \/ - 1 sin l) e = cos 9 + \/ - 1 sin 9.

The definition does not differ from that of cos 9 and sin 9 in geometry,
and this equation is an a priori property of these functions, deducible im-

184 PROFESSOR DE MORGAN, ON THE

mediately from the definition, in any system which gives meaning to \/— 1
from its commencement.

The hardest and most delicate part of this investigation is the connexion
of € 6v/=I with a unit inclined at an angle 9 ; or generally to show that
the operation ( )^~ l changes the exponent of length into one of direction,
and vice versa, without the necessity of inferring this from interpretation.
If we assume beforehand that e^- 1 is real, under the extended definitions,
it would be difficult to imagine what other office ( ) v '- i could perform ;
but such an assumption would not be a proper one, since all the associations
of preceding algebra would lead us to suppose that each extension removes
only one class of inexplicables, and leaves, or perhaps introduces, others. I
cannot complete this part of the subject satisfactorily, but the following
considerations will show that the most simple mode of attaining, upon
an explanation, the technical end of the operation ( ^^ is precisely that
which answers to the above.

Required an operation which repeated n times upon a function of
n quantities shall end by changing the sign of all. Take four quantities,
a, b, c, and d. Successive changes of sign made upon one after the other
will be really different successive operations ; but if we change the sign
of a given one, say the first, and at the same time make a set of periodic
interchanges, writing b for a, c for b, d for c, and a for d, we shall have
an operation which repeated four times will produce the desired effect.
Thus we have successively,

<p(b, c, d, - a), cp(c, d, - a, - b), <p{d, - a - b, - c), <p{- a, - b, - c, - d).

Thus we see in the succession (log a, 9), (-9, log a), (-log a, -9) a
method of passing from A to A' 1 at two similar steps, which does not
involve the use of y/-\. We see the same in (log or, 9), (9, - log a),
and ( - log a, - 9). If then we assume, as a suggestion,

(log a, 9)^ = ( - 9, log a), (log a, 9) ' ^ = (0, - log a),

we find, making A = (log a, 9), the following equations ;

^v=J)^ = A~\ {A-^)'^~ l = A~\ (A^)^- 1 = A,

i i_ _i_

(A^if' 1 = A'\ (A-^k)^ = A~ t , (A^M)'^- 1 = A,

FOUNDATION OF ALGEBRA. 185

in perfect fulfilment of all the fundamental conditions which prescribed
definitions impose. The assumption gives

(aE e )^~ l = J27V=i , 6 iog«.v=r 5

where E e ^~ l must be a symbol of length, and e lo s<>V=i of a unit inclined
at the angle log a. Consequently e 6 ^ 1 must signify a unit inclined at an
angle 9.

It might be asked whether there is anything in the preceding process
which restricts us to the use of the base e rather than any other, I answer,
nothing whatever : but at the same time there is nothing which binds
us to the use of any particular method of measuring angles. It may be
deduced from the preceding that the base e must be used co-ordinately
with that mode of measurement which I call theoretical*. This connexion
depends entirely upon the purely numerical process by which the equation
6 2irv/=5 = 1 is proved to be satisfied when e and ■*■ have their usual meanings.
If for any reason we prefer the base a, the measure of two right angles
must be tx {log e to the base a\.

I think it cannot be disputed that interpretation should be avoided
where explanation can be given. If where the latter cannot be obtained
suggestion upon such analogies as present themselves were to take its
place, the former would be also replaced by verification. In the present
instance, the attainment of

6 <V=i = cos 9 + </~-\ sin 9 from E 6 = cos 9 + \/^l sin 9

is the verification.

1 now come to the theory of logarithms. It is a circumstance which
I hold to be not a little remarkable, that the ancient form of algebra was
only saved from being convicted of incapacity to produce its own legitimate
results, but very little time before such an escape would have been
rendered impossible by its receiving the necessary accession from the more
extended form. Mr Graves has admitted that his view of the new
logarithms should rather have been that of an extension imperatively

* In those works on Trigonometry which use the arc and angle indiscriminately, this mode
of measurement is said to be in parts of the radius. A term is much wanted which shall not
imply this confusion between arcs and angles ; and I propose that the angle which subtends an
arc equal to the radius shall be called the theoretical unit.
Vol. VII. Part II. A A

186 PROFESSOR DE MORGAN, ON THE

required than of a correction to already existing formulae : and in this
view I perfectly agree. If we define log x, or rather Xx, (reserving log x
for the numerical logarithm of the length) to be any legitimate solution
of e Xr a x, it is plain that the logarithm of n inclined at an angle v, (or
of N) to the base b inclined at an angle (S, (or B) is to be derived (avoid-
ing ambiguity) from

. at logn + v\/ - 1

or \ B N= = ,-^-r — r, J - •

logi + ^-l

This result is real when . ° , = g ; nor is it more surprising that an

impossible quantity (hitherto so called) should have a possible logarithm,
than that exponential operations not containing v - 1, or not inter-
changing exponents of length and direction, should in certain cases enable
us to pass from one line to another. I need not enter into details of the
properties of the preceding equation. If we admit all symbols to be
algebraical (in the old sense) which denote lines drawn in the unit
line or its continuation, whatever may be the number of complete revolu-
tions after which they rest there, we must then admit that the logarithm
of a unit which is in its position for the (m + l) ,h time, with respect to
e which is in its position for the (« + l) th time is

1 + 2»7T\/^1

the form proposed by Mr Graves.

In a work of M. Cauchy, and perhaps in other writings which I am not
acquainted with, mention is made of a singular point in curves which
he calls point d 'arret, at which the branch suddenly stops. Such a point
has long been admitted in the spiral of Archimedes and other curves,
owing to the neglect of making those extensions with regard to the sign of
the radius vector which were necessary to complete the connexion* of polar
and rectangular co-ordinates ; and from the assumption of the impossibility

* On this subject I may be allowed to refer to page 341 of my Treatise on the Differential
Calculus.

FOUNDATION OF ALGEBRA. 187

of which (I speak from memory) D'Alembert drew those instances
in which he contended that the negative quantity is not always the
contrary of the positive quantity. Disregarding such points d'arrit, there
is another sort which frequently occurs (but only in exponential or
logarithmic curves), in which the abruptness of the termination is better
marked. Thus in y = (1 - x) log (1 - x), there is, in our present system,
an absolute cessation of the curve when x = I and y = 0. Here, when
the requisite extensions of the logarithmic theory are made, it will be
seen that there is not an absolute abrupt termination, but the commence-
ment of what French writers have called a branche pointilUe, a part of
a curve, which I do not remember to have seen mentioned in any English
work, except Professor Peacock's Report.

A. DE MORGAN.

University College, London,
October 16, 1839-

a A2

XI. On the Effect of the Non-Residence of Landlords, &f. on the Wealth
of a Community. By J. Tozer, Esq. M.A. Caius College.

[Read March 16, 1840.]

The investigations that have been made by political economists of
the effects produced on the wealth of a community by the non-residence
of its proprietors, have frequently been asserted to furnish results which
are not confirmed by observation. The following is believed to be a
more careful investigation of the problem than any that has yet been
made, and one that accounts for the apparent discrepancy.

While the proprietor resides, his income, subject to such deductions
as are made by direct taxation, will be expended either in purchasing
commodities or in paying for services. Those whose services he retains
will expend what they receive in the same manner, and therefore
the whole income of the proprietor will be expended, either directly or
indirectly, in the purchase of commodities. The necessary and sufficient
division of these will be into two classes, those which have been produced
by the labour and capital of the countrymen of the proprietor, and those
which have been produced by the labour and capital of foreigners, and
which have been placed within his reach by the employment of capital
and labour which may have been either native or foreign.

Of taxes, we need only consider those which the proprietor is con-
strained to pay while he is resident, and whose payment he evades by
non-residence. We may also, without affecting the result, assume these
to be paid when the income is realized, and not before.

We have then, while the proprietor resides, a portion of capital
employed in raising the produce of his estates ; a second portion in raising
such other native commodities as are consumed by himself or those

190 Mr TOZER, ON THE EFFECT OF THE

whose services he retains, and in distributing such foreign productions
as are so consumed; and a third portion, which is commercial capital,
belonging either to resident natives or to foreigners, and which is employed
in purchasing and importing those foreign productions.

Each of these classes is susceptible of the usual further divisions
into fixed and floating.

Call these portions of capital C t C,', C 2 C 2 , C 3 C 3 respectively ; CV C 2 C 3
being fixed, and C X C 2 C 3 floating.

Then if q be the amount of a unit of capital with its profit, C v C 2 , C»
must at the end of the year, or other period of return, yield q C„ q C 2 , q C 3 ,
respectively, and C/, C», C 3 ' must yield annuities which pay the profits
(q — 1)CY, (q — l)C 2 ', (q — 1)C S ', and replace such portions of those capitals
as have been destroyed. Call these annuities A if A 2 , A s , respectively.

Then if q . Q + A x — q(c x + c 2 + ... c„) = q^Lc,

where c„ c 2 ... are portions of capital which respectively yield the returns
r u r 2 ...r„ for each unit, the fixed capital being expressed in terms of
its value as floating capital, and r u r 2 ...r n being a decreasing series, the
proprietor's income will be

l} n c(r — q), which suppose = R;
%l indicating the sum of the terms c, (n - q) c % (r 2 - q)...c„ (r„ - q).

Of this, a part Bt suppose will be paid in those direct taxes the pay-
ment of which will be evaded by non-residence.

A part qC 2 + A 2 will be received by the owners of C 2 + C 2 .

The remainder q.C 3 + A 3 by the owners of C 3 + C 3 . Let this re-
mainder = 2jo ; p„ p 2) &c. being portions of it which are exchanged for
portions of foreign produce of different kinds.

Suppose now H to be the country in which the proprietor's lands
are situated, K that in which the equivalent for p, is produced ; and let
a-, be the price of that equivalent in K.

Then if there be direct intercourse between H and K, and if K
import the produce of the proprietor's estates, this price in H must
have been raised to <r(l + e + v) where ea- is the expence of making

NON-RESIDENCE OF LANDLORDS. 191

the transfer, and >/<r the import duty in H, it being supposed that there
are no export duties. If K do not import the produce of the proprietor's
estates, she will either import some other articles produced in H, or the
produce of some other country which does import from H; and there will
be a series of exchanges effected by equivalents that are produced by
capital other than that we are considering, the last operation of im-
porting to H yielding a tax to- to its revenue.

Hence while the proprietor remains at home the produce of his estate
is thus distributed : —

To native capitalists and labourers, 2^c{r(l — i)+qt) — 2/u, of which
C, + C 8 + C\ + C\+ Ai + A-i- (C'i + C'z) . q is employed in replacing capital.

To revenues of foreign states, 2<r2>?.

To revenue of H, fZ\c(r — q) + t2<t.

To commercial capitalists and labourers, 2o-2e; the remainder of
2/0 replacing the commercial capital with which the foreign purchases
were made.

When the proprietor becomes non-resident the capital C 2 + C' 2 will
be disengaged, because his absence destroys the demand on which its
employment depended ; but a new demand for such commodities as
can be exported with advantage will be created by the absence, because
the rent of the proprietor must now be exported. It is therefore in
raising such commodities that the disengaged capital will be employed.
If then the exports of H be made in manufactured goods, there will
not in general be any variation in the rate of profit, because the employ-
ment of additional capital in manufacturing does not diminish profits, if
the demand for manufactures be proportionably increased.

The income to support the inhabitants of H is therefore precisely
the same before as after the removal ; but there is this difference, the
equivalent for the produce of C 2 + C" 8 was then possessed by the proprietor,
it has now to be created by the labour of those who produce the
commodities, by the exportation of which his rent is paid.

Again, let the exports of H consist of raw produce, then to the
series Ci c 2 ...c n there will be in general added the terms c u+ xC„ +2 ... c„ +m ,

192 Mr TOZER, ON THE EFFECT OF THE

producing respectively the returns r n+1 ... r n + m , an effect which will not
be confined to the estate of the proprietor who has become an absentee.
The rate of profit will be lowered from q - 1 to q' — 1, where q' is
determined by the least return which is now made by an unit of capital ;
the whole rental of H will be increased, and the income of the proprietor
will be raised from

R = X<r -q), to R, = V n+m c(r - q)

whence R-R = *•«; c(r-q) + (q - q')Xc,

which will be the gain of the proprietor. The treasury of H will lose
t2 l „c(r — q) + xS<r. Foreign treasuries will gain R 2>/ — 2/)2>7, Rm,
&c, being the import duties successively paid on R and its equivalents.

If the same amount of commercial capital be required to export
R as was required to export Sp, and import its equivalent, C 3 + C' 3 will
be unaffected ; but if a different amount, and if the owners of this capital
be residents of H, then a portion of C 3 + C" 3 may be diverted to the
same employments as C 2 + C 2 have been compelled to seek ; or a portion
of C»+ C" 2 may be employed as commercial capital. The effects of non-
residence will thus be increased or diminished in degree, but will continue
to be the same in kind.

The distinction between a country which exports manufactures and
one that exports raw produce is not a necessary one, though it may
generally exist. The accurate enunciation of the result appears to be,
that beyond the loss to the revenue the absenteeism of proprietors can
only impair the resources of a community when it forces capital from
a more to a less profitable employment.

As regards the effect on F, the country to which the proprietor has
removed, the nature of the products in which his rent will be imported
is independent of the nature of those he consumes, as well as of those
in which it was produced. The presence of an individual who is without
capital, but who is entitled to an income, will therefore create a demand for
the employment of capital in producing the commodities he requires.

This capital will be drawn from investments where its employment
produces the least return. There will be a gain in direct taxes paid by

NON-RESIDENCE OF LANDLORDS. ' 19&

the immigrant, and beyond this there will be a gain, if supplying his wants
afford a more profitable employment for capital than that from "which it
is withdrawn. This will not necessarily be the case, since the importation
of the income may have rendered some species of home production un-
profitable, and the demand for capital being thus accompanied by its
disengagement, the rate of profit may be unaffected. In the case in
which F imports raw produce, there will be an abstraction of capital
from its production at home.

If then c x c 2 ...c„ be portions of capital, the units of which yielded
before the withdrawal r x r % ...r v respectively of produce, r, r 2 ...r v being
a decreasing series, and if c^ +J ... c„ be withdrawn, the rate of profit being
changed from q to q x , the whole rental of F will be reduced from

T v c(r-q), to ^c(r-q l ),

and therefore diminished by

'*?*€(* - q) + (<?' - q)%e.

If the imports of F be made in manufactures, there will not in general
be any increase of wealth consequent on the residence in F of the pro-
prietor whose home is H, beyond the direct taxes he pays; but there
will be a substitution of a portion of income obtained without labour for
an equal portion obtained by labour.

The reason why these results contradict general opinion on the subject
is this : we are compelled, as preliminary to our investigations, to limit
by definition the meaning of the terms we employ. The proprietor,
in the vocabulary of the political economist, is simply the individual
entitled to a certain income when it can be realized, and constantly
either anticipating that income, or devoting it when received to the
purchase of products whose creation rendered the employment of the
capital of others necessary ; his absence therefore does not involve any
removal of capital, and consequently does not diminish the means of
supporting human life. It has also in this investigation been supposed
that he and his dependents expend the whole income to which he is
entitled, without leaving any trace of that expenditure in the shape
of capital accumulated, with a view either to durable or prospective
improvement, or to profit. This supposition must have been tacitly

Vol. VII. Part II. BB

19* Mr TOZER, ON THE EFFECT OF THE

made whenever the wealth of a country has been pronounced to be un-
affected by the non-residence of its landlords.

Let us now suppose a portion of the proprietor's income to have been,
by himself or by his dependents, either accumulated or itself employed
as capital in producing the commodities for which it is exchanged. Call
this portion a, and let a part of it (1 - /) a be expended without accumu-
lation, and a part la be employed as capital with a view to profit. If then
the rate of profit continued constant, and the whole of the proceeds of
la were employed as capital, we should have had in the x + 1 th year
to expend on labour, instead of a,

"{ l f=T +{1 - l) }-

and the fund for employing labour will therefore have been at the
commencement of this year diminished by qla.± — .

Or, if we suppose the rate of accumulation in any one year to be
changed to a v th part of what it was in the preceding, from an alteration
in the rate of profit, or from a different proportional part of that which
is produced being accumulated ; the fund for employing labour will be
at the end of the first, second,.... years, instead of

a, a{l + lq\, a\\ + l(q + q 2 )}, &c a, «(1 + Iqv), a \1 + l(qv 2 + ^V" 1 " 2 )!, &c. ;
and the loss to this fund during the x + 1 th year would be

a{l + llL(qv 2 )*'*}, y to be taken from x - 1 to 1.

This expression may be made to include the loss of the proprietor's
services as a skilled labourer.

But further, a consequence of the absence of the proprietor may be the
removal of a portion of the capital which that absence has forced into new
investments, and the destruction of another portion. Of the capital C 2 + C 2 \
then, let the part mC 2 + n C 2 ' be removed without a change of residence of the
owners, and the part m'C 2 + n'd' be either destroyed or removed in such
a way, that its profit is no longer expended in H ; and let the fractional
parts X, X' respectively of these have been employed in producing capital,
the remainder having been expended without accumulation.

NON-RESIDENCE OF LANDLORDS. 195

Then the resultant loss to the community from the latter part will be
(1 -X)\mC i + n [A, -(q-l) C,']} + (1 - X') (m'C 2 q + n'.A,) ;
and from the former part, at beginning of 1st year,

XmQ + X'm'C 2 ;
at beginning of 2nd year,

XmC % . q + X'm'Q . q + \n {A 2 - (q - 1) C s '} + X'ra,^ ;
at beginning of 3rd year,

XmCtfv + X'm'Crfv + Xn {A,q - {qv - 1) Q'q] + X'n^qvi

x-i

at beginning of x + 1 th year,

{(Xm + X'm') Qqif-* + (Xn + X'n'v*- l )A 2 - Xn (qtf- 1 - 1) C^\ {qv~)*-K
Hence, during the #+l th year, the whole possible loss of income will
be: —

From expenditure of Proprietor, «{1 + l^ =x . x (qv * )"-"}.

From Capital removed or destroyed whose profits would have been
expended without accumulation,
(m + m'q) C 2 — n (q — 1) C 8 ' + {n + n) A 2 + X\n(q - 1) C 2 ' -mC t - hA^

- X' {m'&q + n'A 2 ).

From Capital removed or destroyed whose profits would have been
accumulated,

x— 2

{X [(mC 2 - n C,') qv*- 1 + n (CV+ A,)] + X'(m'C 2 q + n x A^v*"\ (qv—)*- 1 .
The greatest possible loss in this year, when v, I, m', n', X', each = 1, and
.•. m, n each = 0, will be

and the least when /, X, X', each = 0,

a + (m + m t q) C 2 - n (q — 1) C/ + (n + n') A r

As far as the destruction of Capital is concerned the investigation, of
course, applies to the case of proprietors becoming absentees, and not of
their continuing so.

The general effects of absenteeism may be thus enunciated. There will
in all cases be a loss to the home-revenue in those direct taxes whose
payment can be evaded by absence. There will, whenever there are
duties on importation and not on exportation, be a further loss of the

BBS

196

Mr TOZER, ON THE NON-RESIDENCE OF LANDLORDS.

import-duty paid on the foreign productions which the proprietor consumed
when at home.

Beyond this there will be a diminution of the aggregate income
of the community whenever the capital that is disengaged by the ab-
senteeism is forced into less profitable employments than those it previously
occupied, and in no other case. When therefore the country from which
the proprietor absents himself exports raw produce, there will generally,
though not necessarily, be a loss beyond the loss to the revenue, and this
loss will be accompanied by a general increase of rental. When it exports
manufactures, there will not in general be any loss beyond that to the
revenue.

There will however in all cases be this further and very important
effect : though the income which the proprietor removes may be replaced,
it must be replaced by labour, and there will therefore be substituted
for the leisure class, which a part of that income maintained, a class
who must by their own exertions produce the incomes on which they
subsist; and there is nothing in the conditions of the problem to limit
the extent to which the subdivision of income may, among the members
of this class, be carried, or to fix the minimum that may be enjoyed
by each.

It is necessary to the truth of these results, that the withdrawal of
the proprietor should cause no removal of capital, that any part of the
proprietor's income which was not expended should still be saved at
home, and that no part should have been consumed without calling
for the employment of capital.

In applying the result to any particular country, the first step is to
decide how far, in the case of its absentees, these conditions are fulfilled ;
if they be not fulfilled, or if the individuals who remove had in any
degree the qualities of productive labourers, the wealth of the community
must be impaired by their absence ; and the injury is capable of increasing
with time to an indefinite extent.

J. TOZER.

Caius College, Cambridge,
March 16, 1840.

XII. Demonstration that all Matter is heavy. By the Rev. William
Whewell, B.D. Fellow of Trinity College and Professor of Moral
Philosophy.

[Read February 22, 184].]

The discussion of the nature of the grounds and proofs of the most
general propositions which the physical sciences include, belongs rather
to Metaphysics than to that course of experimental and mathematical
investigation by which the sciences are formed. But such discussions seem
by no means unfitted to occupy the attention of the cultivators of physical
science. The ideal, as well as the experimental side of our knowledge
must be carefully studied and scrutinized, in order that its true import may
be seen ; and this province of human speculation has been perhaps of late
unjustly depreciated and neglected by men of science. Yet it can be
prosecuted in the most advantageous manner by them only : for no one can
speculate securely and rightly respecting the nature and proofs of the truths
of science without a steady possession of some large and solid portions of
such truths. A man must be a mathematician, a mechanical philosopher,
a natural historian, in order that he may philosophize well concerning
mathematics, and mechanics, and natural history ; and the mere metaphy-
sician who without such preparation and fitness sets himself to determine
the grounds of mathematical or mechanical truths, or the principles of
classification, will be liable to be led into error at every step. He must
speculate by means of general terms, which he will not be able to use as
instruments of discovering and conveying philosophical truth, because he
cannot, in his own mind, habitually and familiarly, embody their import in
special examples.

Acting upon such views, I have already laid before the Philosophical
Society of Cambridge essays on such subjects as I here refer to ; especially a

198 PROFESSOR WHEWELL's DEMONSTRATION

memoir " On the Nature of the Truth of the Laws of Motion," which was
printed by the Society in its Transactions. This memoir appears to have
excited in other places, notice of such a kind as to shew that the minds of
many speculative persons are ready for and inclined towards the discussion
of such questions. I am therefore the more willing to bring under con-
sideration another subject of a kind closely related to the one just men-
tioned.

The general questions which all such discussions suggest, are (in the
existing phase of English philosophy) whether certain proposed scientific
truths, (as the laws of motion,) be necessary truths ; and if they are neces-
sary, (which 1 have attempted to shew that in a certain sense they are,) on
what ground their necessity rests. These questions may be discussed in a
general form, as I have elsewhere attempted to shew. But it may be
instructive also to follow the general arguments into the form which they
assume in special cases ; and to exhibit, in a distinct shape, the incongrui-
ties into which the opposite false doctrine leads us, when applied to par-
ticular examples. This accordingly is what I propose to do in the present
memoir, with regard to the proposition stated at the head of this paper,
namely, that all matter is heavy.

At first sight it may appear a doctrine altogether untenable to assert
that this proposition is a necessary truth : for, it may be urged, we have no
difficulty in conceiving matter which is not heavy ; so that matter without
weight is a conception not inconsistent with itself; which it must be if the
reverse were a necessary truth. It may be added, that the possibility of
conceiving matter without weight was shewn in the controversy which
ended in the downfall of the phlogiston theory of chemical composition ;
for some of the reasoners on this subject asserted phlogiston to be a body
with positive levity instead of gravity, which hypothesis, however false,
shews that such a supposition is possible. Again, it may be said that
weight and inertia are two separate properties of matter : that mathemati-
cians measure the quantity of matter by the inertia, and that we learn by
experiment only that the weight is proportional to the inertia ; Newton's
experiments with pendidums of different materials having been made with
this very object.

THAT ALL MATTER IS HEAVY. 199

I proceed to reply to these arguments. And first, as to the possibility of
conceiving matter without weight, and the argument thence deduced, that
the universal gravity of matter is not a necessary truth, I remark, that it is
indeed just, to say that we cannot even distinctly conceive the contrary of
a necessary truth to be true; but that this impossibility can be asserted only
of those perfectly distinct conceptions which result from a complete deve-
lopement of the fundamental idea and its consequences. Till we reach this
stage of developement, the obscurity and indistinctness may prevent our
perceiving absolute contradictions, though they exist. We have abundant
store of examples of this, even in geometry and arithmetic; where the
truths are universally allowed to be necessary, and where the relations which
are impossible, are also inconceivable, that is, not conceivable distinctly. Such
relations, though not distinctly conceivable, still often appear conceivable
and possible, owing to the indistinctness of our ideas. Who, at the first
outset of his geometrical studies, sees any impossibility in supposing the
side and the diagonal of a square to have a common measure? Yet they
can be rigorously proved to be incommensurable, and therefore the attempt
distinctly to conceive a common measure of them must fail. The attempts
at the geometrical duplication of the cube, and the supposed solutions, (as
that of Hobbes) have involved absolute contradictions ; yet this has not
prevented their being long and obstinately entertained by men, even of
minds acute and clear in other respects. And the same might be shewn to
be the case in arithmetic. It is plain, therefore, that we cannot, from the
supposed possibility of conceiving matter without weight, infer that the
contrary may not be a necessary truth.

Our power of judging, from the compatibility or incompatibility of our
conceptions, whether certain propositions respecting the relations of ideas
are true or not, must depend entirely, as I have said, upon the degree of
developement which such ideas have undergone in our minds. Some of
the relations of our conceptions on any subject are evident upon the first
steady contemplation of the fundamental idea by a sound mind : these are
the axioms of the subject. Other propositions may be deduced from the
axioms by strict logical reasoning. These propositions are no less necessary
than the axioms, though to common minds their evidence is very different.
Yet as we become familiar with the steps by which these ulterior truths are

200 PROFESSOR WHEWELL's DEMONSTRATION

deduced from the axioms, their truth also becomes evident, and the contrary
becomes inconceivable. When a person has familiarized himself with the
first twenty-six propositions of Euclid, and not till then, it becomes evident
to him, that parallelograms on the same base and between the same parallels
are equal ; and he cannot even conceive the contrary. When he has a little
further cultivated his geometrical powers, the equality of the square on the
hypothenuse of a right-angled triangle to the squares on the sides, becomes
also evident ; the steps by Avhich it is demonstrated being so familiar to the
mind as to be apprehended without a conscious act. And thus, the contrary
of a necessary truth cannot be distinctly conceived ; but the incapacity of
forming such a conception is a condition which depends upon cultivation,
being intimately connected with the power of rapidly and clearly perceiving
the connection of the necessary truth under consideration with the elemen-
tary principles on which it depends. And thus, again, it may be that there
is an absolute impossibility of conceiving matter without weight ; but then,
this impossibility may not be apparent, till we have traced our fundamental
conceptions of matter into some of their consequences.

The question then occurs, whether we can, by any steps of reasoning,
point out an inconsistency in the conception of matter without weight.
This I conceive we may do, and this I shall attempt to shew.

The general mode of stating the argument is this : — the quantity of
matter is measured by those sensible properties of matter which undergo
quantitative addition, subtraction and division, as the matter is added, sub-
tracted and divided. The quantity of matter cannot be known in any other
way. But this mode of measuring the quantity of matter, in order to be
true at all, must be universally true. If it were only partially true, the
limits within which it is to be applied would be arbitrary ; and therefore
the whole procedure would be arbitrary, and, as a method of obtaining
philosophical truth, altogether futile.

We may unfold this argument further. Let the contrary be sup-
posed, of that which we assert to be true : namely, let it be supposed that
while all other kinds of matter are heavy, (and of course heavy in propor-
tion to the quantity of matter) there is one kind of matter which is abso-
lutely destitute of weight; as, for instance, phlogiston, or any other element.

THAT ALL MATTER IS HEAVY. 201

Then where this weightless element (as we may term it) is mixed with
weighty elements, we shall have a compound, in which the weight is no
longer proportional to the quantity of matter. If, for example, 2 measures of
heavy matter unite with 1 measure of phlogiston, the weight is as 2, and
the quantity of matter as 3. In all such cases, therefore, the weight ceases
to be the measure of the quantity of matter. And as the proportion of the
weighty and the weightless matter may vary in innumerable degrees in
such compounds, the weight affords no criterion at all of the quantity of
matter in them. And the smallest admixture of the weightless element is
sufficient to prevent the weight from being taken as the measure of the
quantity of matter.

But on this hypothesis, how are we to distinguish such compounds from
bodies consisting purely of heavy matter ? How are we to satisfy ourselves
that there is not, in every body, some admixture, small or great, of the
weightless element ? If we call this element phlogiston, how shall we know
that the bodies with which we have to do are, any of them, absolutely free
from phlogiston ?

We cannot refer to the weight for any such assurance; for by supposition
the presence and absence of phlogiston makes no difference in the weight.
Nor can any other properties secure us at least from a very small admixture;
for to assert that a mixture of 1 in 100 or 1 in 10 of phlogiston would
always manifest itself in the properties of the body, must be an arbitrary
procedure, till we have proved this assertion by experiment : and we cannot
do this till we have learnt some mode of measuring the quantities of matter
in bodies and parts of bodies ; which is exactly what we question the possi-
bility of, in the present hypothesis.

Thus, if we assume the existence of an element, phlogiston, devoid of
weight, we cannot be sure that every body does not contain some portion of
this element ; while we see that if there be an admixture of such an element,
the weight is no longer any criterion of the quantity of matter. And thus
we have proved, that if there be any kind of matter which is not heavy, the
weight can no longer avail us, in any case or to any extent, as a measure of
the quantity of matter.

Vol. VII. Part II. CC

202 PROFESSOR WHEWELL's DEMONSTRATION

I may remark, that the same conclusion is easily extended to the case in
which phlogiston is supposed to have absolute levity ; for in that case, a
certain mixture of phlogiston and of heavy matter would have no weight,
and might be substituted for phlogiston in the preceding reasoning.

I may remark, also, that the same conclusion would follow by the same
reasoning, if any kind of matter, instead of being void of weight, were
heavy, indeed, but not so heavy, in proportion to its quantity of matter, as
other kinds.

On all these hypotheses there would be no possibility of measuring quan-
tity of matter by weight at all, in any case, or to any extent.

But it may be urged, that we have not yet reduced the hypothesis of
matter without weight to a contradiction ; for that mathematicians measure
quantity of matter, not by weight, but by the other property, of which we
have spoken, inertia.

To this I reply, that, practically speaking, quantity of matter is always
measured by weight, both by mechanicians and chemists : and as we have
proved that this procedure is utterly insecure in all cases, on the hypothesis
of weightless matter, the practice rests upon a conviction that the hypo-
thesis is false. And yet the practice is universal. Every experimenter
measures quantity of matter by the balance. No one has ever thought of
measuring quantity of matter by its inertia practically : no one has con-
structed a measure of quantity of matter in which the matter produces its
indications of quantity by its motion. When we have to take into account
the inertia of a body, we inquire what its weight is, and assume this as the
measure of the inertia; but we never take the contrary course, and ascertain
the inertia first in order to determine by that means the weight.

But it may be asked, Is it not then true, and an important scientific
truth, that the quantity of matter is measured by the inertia ? Is it not true,
and proved by experiment, that the weight is proportional to the inertia f
If this be not the result of Newton's experiments mentioned above, what, it
may be demanded, do they prove ?

To these questions I reply : It is true that quantity of matter is mea-
sured by the inertia, for it is true that inertia is as the quantity of matter.

THAT ALL MATTER IS HEAVY. 203

This truth is indeed one of the laws of motion. That weight is propor-
tional to inertia is proved by experiment, as far as the laws of motion are
so proved : and Newton's experiments prove one of the laws of motion, so
far as any experiments can prove them, or are needed to prove them.

That inertia is proportional to weight, is a law equivalent to that law
which asserts, that when pressure produces motion in a given body, the
velocity produced in a given time is as the pressure. For if the velocity be
as the pressure, when the body is given, the velocity will be constant if the
inertia also be as the pressure. For the inertia is understood to be that pro-
perty of bodies to which, ceteris paribus, the velocity impressed is inversely
proportional. One body has twice as much inertia as another, if, when the
same force acts upon it for the same time, it acquires but half the velocity.
This is the fundamental conception of inertia.

In Newton's pendulum experiments, the pressure producing motion was
a certain resolved part of the weight, and was proportional to the weight.
It appeared by the experiments, that whatever were the material of which
the pendulum was formed, the rate of oscillation was the same; that is, the
velocity acquired was the same. Hence the inertia of the different bodies
must have been in each case as the weight : and thus this assertion is true of
all different kinds of bodies.

Thus it appears that the assertion, that inertia is universally proportional
to weight, is equivalent to the law of motion, that the velocity is as the
pressure. The conception of inertia (of which, as we have said, the funda-
mental conception is, that the velocity impressed is inversely proportional
to the inertia,) connects the two propositions so as to make them identical.

Hence our argument with regard to the universal gravity of matter
brings us to the above law of motion, and is proved by Newton's experiments
in the same sense in which that law of motion is so proved.

Perhaps some persons might conceive that the identity of weight and
inertia is obvious at once ; for both are merely resistance to motion ; — inertia,
resistance to all motion (or change of motion) — weight, resistance to motion
upwards.

But there is a difference in these two kinds of resistance to motion. Inertia
is instantaneous, weight is continuous resistance. Any momentary impulse

c C2

204 PROFESSOR WHEWELL's DEMONSTRATION

which acts upon a free body overcomes its inertia, for it changes its motion ;
and this change once effected, the inertia opposes any return to the former
condition, as well as any additional change. The inertia is thus overcome by
a momentary force. But the weight can only be overcome by a continuous
force like itself. If an impulse act in opposition to the weight, it may for a
moment neutralize or overcome the weight ; but if it be not continued, the
weight resumes its effect, and restores the condition which existed before
the impulse acted.

But weight not only produces rest, when it is resisted, but motion, when
it is not resisted. Weight is measured by the reaction which would balance
it; but when unbalanced, it produces motion, and the velocity of this
motion increases constantly. Now what determines the velocity thus pro-
duced in a given time, or its rate of increase ? What determines it to have
one magnitude rather than another ? To this we must evidently reply, the
inertia. When weight produces motion, the inertia is the reaction which
makes the motion determinate. The accumulated motion produced by the
action of unbalanced weight is as determinate a condition as the equili-
brium produced by balanced weight. In both cases the condition of the
body acted on is determined by the opposition of the action and reaction.

Hence inertia is the reaction which opposes the weight, when unba-
lanced. But by the conception of action and reaction, (as mutually deter-
mining and determined,) they are measured by each other ; and hence the
inertia is necessarily proportional to the weight.

But when we have reached this conclusion, the original objection may
be again urged against it. It may be said, that there must be some fallacy
in this reasoning, for it proves a state of things to be necessary when
we can so easily conceive a contrary state of things. Is it denied, the
opponent may ask, that we can readily imagine a state of things in
which bodies have no weight? Is not the uniform tendency of all bodies
in the same direction not only not necessary, but not even true? For
they do in reality tend, not with equal forces in parallel lines, but to
a center with unequal forces, according to their position : and we can
conceive these differences of intensity and direction in the force to be

THAT ALL MATTER IS HEAVY. 205

greater than they really are; and can with equal ease suppose the force
to disappear altogether.

To this I reply, that certainly we may conceive the weight of bodies
to vary in intensity and direction, and by an additional effort of imagi-
nation, may conceive the weight to vanish : but that in all these sup-
positions, even in the extreme one, we must suppose the rule to be universal.
If any bodies have weight, all bodies must have weight. If the direction
of weight be different in different points, this direction must still vary
according to the law of continuity ; and the same is true of the intensity
of the Aveight. For if this were not so, the rest and motion, the velocity
and direction, the permanence and change of bodies, as to their mechanical
condition, would be arbitrary and incoherent: they would not be subject
to mechanical ideas ; that is, not to ideas at all : and hence these conditions
of objects would in fact be inconceivable. In order that the universe
may be possible, that is, may fall under the conditions of intelligible
conceptions, we must be able to conceive a body at rest. But the rest of
bodies (except in the absolute negation of all force) implies the equilibrium
of opposite forces. And one of these opposite forces must be a general
force, as weight, in order that the universe may be governed by general
conditions. And this general force, by the conception of force, may
produce motion, as well as equilibrium ; and this motion again must
be determined, and determined by general conditions; which cannot be,
except the communication of motion be regulated by an inertia propor-
tional to the weight.

But it will be asked, Is it then pretended that Newton's experiment,
by which it was intended to prove inertia proportional to weight, does
really prove nothing but what may be demonstrated a priori ? Could
we know, without experiment, that all bodies, — gold, iron, wood, cork, —
have inertia proportional to their weight? And to this we reply, that
experiment holds the same place in the establishment of this, as of the
other fundamental doctrines of mechanics. Intercourse with the external
world is requisite for developing our ideas; measurement of phenomena
is needed to fix our conceptions and to render them precise: but the
result of our experimental studies is, that we reach a position in which
our convictions do not rest upon experiment. We learn by observation

206 PROFESSOR WHEWELL's DEMONSTRATION

truths of which we afterwards see the necessity. This is the case with
the laws of motion, as I have repeatedly endeavoured to shew. The same
will appear to be the case with the proposition, that bodies of different
kinds have their inertia proportional to their weight.

For bodies of the same kind have their inertia proportional to their
weight, both quantities being proportional to the quantity of matter.
And if we compress the same quantity of matter into half the space,
neither the weight nor the inertia is altered, because these depend on
the quantity of matter alone. But in this way we obtain a body of
twice the density ; and in the same manner we obtain a body of any other
density. Therefore whatever be the density, the inertia is proportional
to the quantity of matter. But the mechanical relations of bodies cannot
depend upon any difference of kind, except a difference of density. For
if we suppose any fundamental difference of mechanical nature in the
particles or component elements of bodies, we are led to the same con-
clusion, of arbitrary, and therefore impossible, results, which we deduced
from this supposition with regard to weight. Therefore all bodies of
different density, and hence, all bodies whatever, must have their inertia
proportional to their weight.

Hence we see, that the propositions, that all bodies are heavy, and that
inertia is proportional to weight, necessarily follow from those fundamental
ideas which we unavoidably employ in all attempts to reason concerning
the mechanical relations of bodies. This conclusion may perhaps appear the
more startling to many, because they have been accustomed to expect that
fundamental ideas and their relations should be self-evident at our first
contemplation of them. This, however, is far from being the case, as I have
already shewn. It is not the first, but the most complete and developed
condition of our conceptions which enables us to see what are axiomatic
truths in each province of human speculation. Our fundamental ideas
are necessary conditions of knowledge, universal forms of intuition,
inherent types of mental developement ; they may even be termed, if
any one chooses, results of connate intellectual tendencies ; but we cannot
term them innate ideas, without calling up a large array of false opinions.
For innate ideas were considered as capable of composition, but by no
means of simplification : as most perfect in their original condition ; as to

THAT ALL MATTER IS HEAVY. 207

be found, if any where, in the most uneducated and most uncultivated
minds ; as the same in all ages, nations, and stages of intellectual culture ;
as capable of being referred to at once, and made the basis of our reasonings,
without any special acuteness or effort : in all which circumstances the
Fundamental Ideas of which we have spoken, are opposed to Innate Ideas
so understood.

I shall not, however, here prosecute this subject. I will only remark,
that Fundamental Ideas, as we view them, are not only not innate, in any
usual or useful sense, but they are not necessarily ultimate elements of our
knowledge. They are the results of our analysis so far as we have yet
prosecuted it ; but they may themselves subsequently be analysed. It may
hereafter appear, that what we have treated as different Fundamental Ideas
have, in fact, a connexion, at some point below the structure which we
erect upon them. For instance, we treat of the mechanical ideas of force,
matter, and the like, as distinct from the idea of substance. Yet the prin-
ciple of measuring the quantity of matter by its weight, which we have
deduced from mechanical ideas, is applied to determine the substances
which enter into the composition of bodies. The idea of substance supplies
the axiom, that the whole quantity of matter of a compound body is equal
to the sum of the quantities of matter of its elements. The mechanical
ideas of force and matter lead us to infer that the quantity both of the
whole and its parts must be measured by their weights. Substance may,
for some purposes, be described as that to which properties belong ; matter
in like manner may be described as that which resists force. The former
involves the Idea of permanent Being ; the latter, the Idea of Causation.
There may be some elevated point of view from which these ideas may
be seen to run together. But even if this be so, it will by no means affect
the validity of reasonings founded upon these notions, when duly deter-
mined and developed. If we once adopt a view of the nature of knowledge
which makes necessary truth possible at all, we need be little embarrassed
by finding how closely connected different necessary truths are ; and how
often, in exploring towards their roots, different branches appear to spring
from the same stem.

W. WHEWELL.
Grange,

Aug. SI, 1840.

XIII. On the Position of the Axes of Optical Elasticity in Crystals
belonging to the Oblique-Prismatic System. By W. H. Miller,
M.A. F.R.S. Fellow and Tutor of St. John's College, and Pro-
fessor of Mineralogy in the University of Cambridge.

[Read March 21, 1836.]

In a Memoir printed in the 5th Volume of the Cambridge Trans-
actions it is stated, that in crystals belonging to the Oblique-Prismatic
System one of the three rectangular axes of optical elasticity was
always found to coincide with that crystallographic axis (Y, Y') which,
in crystals of this system, is perpendicular to the other two: but that
the positions of the other axes of optical elasticity (££', ££') had no
known relation to the form of the crystal. In some oblique-prismatic
crystals, however, it was found that one of the axes of optical elasticity
f £'» £T was a l so the axis of a principal zone. In the crystals which
I have examined since the publication of the paper already alluded to,
by the same method, this coincidence is found to occur less frequently.
Upon the whole, however, there seems to be no reason for supposing
it accidental in the instances (five or six out of twenty) in which it
has been observed ; but rather that it is a particular case of some
general law connecting the form and optical properties of crystals, in
the discovery of which it is hoped the observations here recorded may
be in some degree instrumental.

The crystals selected for examination are taken principally from
among those which have been described by Mr Brooke in the Annals
of Philosophy for 1823 and 1824. The mutual inclination of two faces
is expressed by the angle between their normals, or the angular distance
of their "poles." An explanation of the notation in which the symbols

Vol. VII. Part II. DD

210

PROFESSOR MILLER ON THE POSITION OF THE AXES

of the simple forms are expressed, and of the method of representing
the form of a crystal by its " sphere of projection," will be found in
the Cambridge Transactions, Vol. V. p. 433. The velocity of light in
air divided by its velocity within the crystal, for a ray in the plane
of the optic axes, and polarized in the same plane, is denoted by ft.
I being the refracting angle of a prism having its edge perpendicular
to the plane of the optic axes, and D the minimum deviation of a ray
refracted through it, polarized in the plane of the optic axes, n sin 1 /
= sin ^ (D + I). The index of refraction of the oil used in some of
the observations is 1.4706 for the brightest rays of the spectrum, a, /3 ;
£, £, denote the extremities of radii of the sphere of projection drawn
parallel to the optic axes and axes of optical elasticity respectively.

(1). In Oxalic Acid, C H 3 , the cleavages being
parallel to the faces m, mm' = 63°. 5', ee' = 34° . 32,
pa = 50°. 40', cp' = 76°. 45', ac m 52°. 35', pm
= 81°. 34', am = 6l°.13',5, cm = 62°.55',5. The
symbols of the simple forms are, p {0 1},
m {1 1 0}, e {Oil}, a {10 1}, c {10 1}.

The apparent directions of the optic axes seen in oil through the faces p
lie in a plane perpendicular to the faces p, e, and make with each other an
angle of 115°. 30'. ^ = 1.499- Hence a/3 m 68°, and the axis of optical
elasticity £ coincides with the axis of the zone peep'.

(2). In Sphene, the faces being denoted by the same letters as in the
treatises of Mohs and Naumann, and the principal cleavages being parallel to
the faces I, y, #/=66 .54', y/=131°.21', fy> = 85°.33',
yx = 21°.5', xp = 39°.19', pql = 85°. 10', pqt = 53°.36',
pqn = 28°. 6'. The symbols of the simple forms are,
q {0 1 0}, p {0 1}, / {1 1 0}, m {1 3 0}, r{0 1 1},
y {1 1}, x {1 02}, o {0 1 3}, t {1 2 1}, d {1 1 3},
n {I 2 3}, u {16 3}, / {T 1 2}, s {14 1}.

The apparent directions of the optic axes seen in water through the
faces x lie in a plane perpendicular to the faces xp, and make angles of

OF OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 211

about 18°. 40', with a normal to the face x. n = 1.631. Hence a/3 = 30°. 22',
and the axis of elasticity £ coincides with the axis of the zone xfq.

(3). In Phosphate of Soda, Na*PH*, according to Mitscherlich
(Annates de Chimie, Tome 19.) ur — 33°. 8', rp =
25°. 24', pf = 50°. 48', fb = 33°. 25', bu = 37°. 17',
dm = 33°. 55', di = 65". 4', dn = 53°. l£, dl = 52°. 9',
dt = 36°. 30', dk = 67°. 6', pt = 6T. 55', pn = 31°. 30',
pm — 73°. 3'. The symbols of the simple forms
are, k {10 0}, d {0 1 0}, p \0 1\, m \1 1 0},
n \1 11}, t {111}, A {201},/ {101}, /• {10 1},
Z {0 2 3}, k {3 1 3}, i {3 1 0}.

The optic axes lie in a plane perpendicular to the faces u, p, f. When
the crystal is immersed in oil, the apparent direction of the optic axis a seen
through the faces p makes an angle of 34°. 30' with a normal to p, and an
angle of 58°. 40' with the apparent direction of the optic axis /3 seen through
artificial surfaces nearly perpendicular to the optic axis (Z. n = 1.40 nearly.
Hence jo a = 36°. 30', jo/3 = 93°. 10', /j£ = 64°.50'. Therefore the axis of
elasticity f very nearly coincides with the axis of the zone rnd. It is not
possible to determine the positions of the optic axes of phosphate of soda
very accurately on account of its feeble double refraction, the imperfection
of its surfaces, and its tendency to effloresce.

(4). In Acetate of Soda, NaAH 6 , ap = 76".25', ph = 68".l&, ha =
35°. 15', mk = 42°. 15', pm = 75°. 35', pf = 420.43, pe = 60\ 22', pg =
81°.8',5. The symbols of the simple forms are, h {1 0}, i {0 10},
p\001\, a {201}, / {111}, e {111}, g {2 2 1}.

When the crystal is immersed in oil, the apparent
directions of the optic axes seen through a slice
bounded by artificial surfaces nearly parallel to the
faces a, make with each other an angle of 62°. 30';
and the apparent direction of the optic axis /3 seen
through artificial sections nearly perpendicular to fi,
makes an angle of 80°. 30', with a normal to p.
m = 1.464. Hence a/3 = 117°.10', p% = 11°.9', «£ = 2°. 26'.

DD 2

212

PROFESSOR MILLER ON THE POSITION OF THE AXES

(5). In Acetate of Oxide of Zinc, ZnAH\ the cleavage being parallel
to the face p, ph = 46°. 30', pc = 79°. 55', "mm' = 67°.24', pm = 67°.33',
pg = 7 '5". 30', g-g-' = 58°. 43'. The symbols of the simple forms are,
c {10 0}, j» {0 01$, h {101}, m |H 1}, g fill}-

The optic axes lie in a plane perpendicular to
the faces p, h, c. The apparent direction of the
optic axis a, seen in air through the faces p,
makes an angle of 50°. 15', with a normal to p.
When the crystal is immersed in oil, the apparent
directions of the optic axes seen through the
faces p, make with each other an angle of 79°. 15'.
P- 1.494. Hence p£= 11°. 16', A£=35°. 14', a/3 = 84°.30'.

(6). In Bicarbonate of Potash, KC 2 H, me =
53°. 15', ra* = 76°.35', mf= 127°.52', rf</'=42°.0'.
The symbols of the simple forms are, m {10 0},
t {0 1}, / {TO 1}, e {2 3}, d \\ 1 0}.

The apparent direction of the optic axis a seen
in air through the faces e, makes an angle of 56°. 45',
with a normal to e. The apparent directions of the optic axes seen in oil
through the faces e make with each other an angle of 83°. /x = 1.482.
Hence ea = 48°. 21', e& = 47°. 53', e% = 6 '.28', a/3 = 81°. 38'.

(7). In Tartaric Acid, TH, mm' = 88°.30', ee'= 76°.30', pm = 97°. 10',
ph = 80°. 3'. The symbols of the simple forms are, h {100}, p {001},
m {1 1 0}, a {1 1}, e {Toi}, e {0 1 1}.

The apparent directions of the optic axes seen in
oil through artificial sections perpendicular to the faces
p, h, lie in a plane inclined at an angle of 69°. 30' to
the face p, and make with each other an angle of 1 03°.
/u= 1.542 nearly. Hence pi = 20°. 30', />£ = 10°. 33',
a/3= 96°. 36'.

OF OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 213

(8). In Pyroxene, the faces being denoted by the same letters
as in the treatises of Mohs and Naumann, the symbols
of the simple forms are, p {00 1}, I {0 10}, r {100},
m{110},/{8 10}, MI 01}, *{0 11}, u\21l\,
as {12 1}, o 1 2 1}, X {2 3 1}.

The optic axes lie in a plane parallel to the
face /. The apparent direction of the optic axis a
seen in air through sections perpendicular to the faces
m, m', makes an angle of 74° with a normal to /. The apparent
direction of the optic axis /3 seen in water through the faces r, r, makes
an angle of 27°, 40', with a normal tor. M = 1.680. Hence ar' = 80°.34',
/3/= 21°. 38', £r'= 56°. 6'.

The crystals of Pyroxene in which I first attempted to determine the
position of the optic axes were all twins composed of individuals of
unequal size, the twin-axis being perpendicular to the face r. Con-
sequently, a slice bounded by planes perpendicular to the faces m, m'
exhibited two systems of rings unequally bright, making with each
other an angle of 32°, which was bisected by the axis of the zone mr.
These rings were erroneously supposed to belong to the same crystal,
till the mistake was pointed out to me by Professor Norrenberg. The
best crystals which I have been able to procure for measurement give
pr = 74°. 20', nearly. In a twin crystal in Mr Brooke's collection,
the face t of one individual coincides accurately with the face p of
the other. This shews that Pyroxene may quantitatively be referred
to the right prismatic system. The position of the optic axes, as well
as the nature of the symmetry of the faces u, ss, o, X, prove, however,
clearly that it belongs to the oblique-prismatic system.

(9). In crystals of Sugar the faces are too uneven to admit of de-
termining the angles they make with each other
nearer than within, perhaps, half a degree of the truth.
a being the face parallel to which a very distinct
cleavage exists, mm =79°.20', ar = ll6°.40', «c = 75 n .30',
nearly. The symbols of the simple forms are, a {10 0},
c {0 1}, r{l 1}, * {1 1}, n {0 1 1}, p {1 1 1}.

214

PROFESSOR MILLER ON THE POSITION OF THE AXES

The apparent directions of the optic axes a, /3, seen in oil through
the faces a, a, make angles of 1°.32' and 49°.58', with a normal to a.
iu = 1.57. Hence a a = 1°.26\ fl/3 = 45°.50', a% = 22°. 12', r^ = 4°.28'.

(10). In Tartrate of Potash, KTH-, the cleavages being parallel to
m, t, me = 37°. 47', et = 52°.42', Z>6' = 45°.20'. The symbols of the
simple forms are, c \0 1 0}, e |0 l\, t \\ 1$,

IB {10 1}, * \0 1 1}.

The apparent directions of the optic axes seen in oil
through the faces t lie in a plane perpendicular to the
face c, making an angle of 67°. 30' with the face /.
They make with each other an angle of 64°. 45', and
therefore angles of 38°. 43', with a normal to t. n = 1.526
nearly. Hence, supposing a ray in the direction of the optic axes to
be refracted in the same manner as at the surface of glass, having 1.526
for its index of refraction, t% m 21°. 20', a/3 = 118° nearly.

The above assumption, though not strictly correct, will not occasion
any considerable error in the present instance. This appears to be the only
practicable method of determining (approximately) the positions of the
optic axes, when the plane in which they lie is not perpendicular to
the faces through which they are seen. It is used in the two following-
cases.

(11). In Chlorate of Potash, KCl, the cleavages being parallel to
the faces m, m', mm' = 104°. 0', ee' = 79°.30', pm = 74°.30'. The sym-
bols of the simple forms are, p J0 1J, m \1 10},
e {0 1 1}, c {I 1}.

The apparent directions of the optic axes seen in oil
through the faces p lie in a plane parallel to the axis of
the zone pc, making an angle of 52° with the face p,
and they make with each other an angle of 28°. 15'.
m = 1.507 nearly. Hence pi = 37°.42', a/3 - 152°.30'
nearly.

OF OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 215

(12). In Sulphate of Soda, Na iSu H w , hp = 72°. 16', pc=W-\5'
km m 40° . 12', kl = 22° . 54', ke = 49° . 54'. The
symbols of the simple forms are, k J01 Oj,
h {100}, p {001}, / {120}, e {Oil},
»i {11 0}.

The apparent directions of the optic axes
seen in oil through the faces h lie in a plane
making an angle of 78°. 30' with the face //, and
make with each other an angle of 97°. 30'.
n = 1.44 nearly. Hence h% = 12°. 24', a/3 = 80°. 26' nearly.

(13.) In Hydrous Oxalate of Lime, Ca C H, a new mineral species
described by Mr Brooke in the Philosophical Magazine for June 1840,
b being the face parallel to which a very distinct cleavage exists,
cm = 50°. 18', cf = 65°. 28, ca = 37°. 24', 5, cu = 31°. 3', cs = 28°. 41',
pm = 76°. 46', pb = 70°. 33', pern = 72°. 41'. The symbols of the
simple forms are, c {0 1 0}, p {0 1}, j» {11 0},
a \0 1 1}, b {1 1}, u {1 2 0}, / {1 1 2}, s \\ 3 2}.

The optic axes could not be seen ; the position of
the axes of elasticity was however determined approx-
imately by placing the crystal in a polarizing apparatus,
having the planes of polarization and analyzation at
right angles to each other, with the face c perpen-
dicular to the axis of the instrument, and observing the position of the
face p when the crystal ceased to transmit light. In this manner it was
found that J| = 8°.

W. H. MILLER.

XIV. On a New Construction of the Going-Fusee. By G. B. Airy, Esq.

Astronomer Royal.

[Read March 2, 1840.]

I should not have presumed to occupy the time of the Cambridge Phi-
losophical Society with a mere description of a mechanical construction,
if I did not conceive that it might possess some interest for them, first,
as a modification of or rather a substitute for a contrivance, whose elegance
and importance have been universally acknowledged, but which fails
(from practical reasons only) in certain cases. And secondly, as an object
of local interest, the only existing application of the new construction
being in the mounting of the magnificent telescope, which the University
owes to the munificence of the Duke of Northumberland.

The object of the going-fusee is, as is well known, to maintain exactly
the same action (whatever its amount may be) upon the first wheel of
a clock, while the clock is being wound up, as while it is going in its
ordinary way : supposing that the time required for winding up is not
very long.

It was invented by Harrison ; and has always appeared to me one of
the most beautiful of the many beautiful contrivances in a highly-
finished time-keeper.

When I was arranging the clock-work for the Northumberland tele-
scope, I soon perceived that it would be necessary to depart from Harrison's
construction in the going-fusee part. This was rendered imperative by
the magnitude of the force which, as it appeared probable, would be re-
quired to maintain the motion of the clock. A strain of lOOlbs. on the
cord was to be provided for : and therefore the remontoir spring must
be strong enough to support lOOlbs. without breaking, yet sufficiently

Vol. VII. Part II. EE

218 Mb AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE.

flexible and elastic to expand without great diminution of that force
through a sensible space. There is doubtless no difficulty in satisfying
these conditions in the case of a coach-spring, or where there is abundance
of room : but, where the spring must be contained within not a clock
but a clock-wheel, there is considerable difficulty. The only way in which
I could hope to use the principle, must be by adopting a barrel, ratchet,
click, and first wheel, exactly as in a kitchen - clock ; and removing the
ratchet-wheel of Harrison's fusee with its click and going-spring to the
spindle of the next wheel, where the forces are much diminished. But
here it would be necessary to use a spring which is coiled several times
round the spindle ; else, as this second wheel revolves more rapidly than
the first, the spring would be too much relaxed before the cessation of the
pressure of the hand allowed the weight to act again. The difficulty of
manufacturing the spring would be great, and in all contrivances re-
quiring a steel spring there was the risk of rust, against which I could not
hope to secure the machinery.

I might have adopted the contrivance known as the endless cord of
Huyghens, which has been employed in Fraunhofer's clocks. The only
objection to the use of this construction for the Northumberland clock
was, that the spikes in the gorge of the pulley, which are necessary to
prevent the cord from slipping, would speedily have torn the cord to
pieces, when a weight of 100 lb. was attached.

Abandoning the spring and the endless rope, my first idea was, to
use a new weight in such a manner as to produce exactly the same effect
and in the same place as Harrison's going-spring. Various constructions
presented themselves ; but those founded on the following principle, ap-
peared the most feasible : — The action of a spring may be exactly imitated
by that of a jointed lozenge : the two parts which are to be connected
by the spring being two opposite angles of the lozenge, and the two
other angles being pulled apart by the action of constant weights. In
the application of this principle, the parts to be connected by the spring
or lozenge would be on the circumference of the barrel and wheel, and
the two other angles would therefore be on the same circumference : but
there was no difficulty in effecting the pulling apart of these angles by

Mr AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE. 219

a force in the axis of the barrel, which, by a proper application of bell-
cranks, could easily be effected by a weight. But the construction pro-
duced a trifling friction in the ordinary going of the clock, and was not
elegant, and I therefore abandoned it.

Finally, I had the good fortune to imagine a construction entirely dif-
ferent, with which, in all respects, I am fully satisfied. It is based upon
the following principle. If a lever abc, fig. 1, is used to produce pressure
at the point c, b being its fulcrum, and a the point at which the force is
applied : then the same effect may be produced on c, by making a the
fulcrum, provided that at b we apply a force exactly equal and opposite to
the pressure which the fulcrum at b sustained when the force was applied
at a.

To apply this to the first wheel of a clock. Suppose (for simplicity of
present consideration and of future construction) the axes of the first wheel
and second wheel to be in the same horizontal plane. Let a fig. 2. be the
point of the barrel from which the weight depends : b its center, c the
point at which the toothed wheel, connected with the barrel, acts upon the
pinion of the next wheel. Then, during the ordinary action of the clock,
abc may be considered as a lever, of which b is the fulcrum, sustaining a
downwards pressure, a the point of application of the force, and c the point
on which it is to produce an effect. Suppose, in the operation of winding
up, the force acting at a to be removed. Then, by the theorem which I
have lately mentioned, the same effect may still be produced on c; pro-
vided that we can so arrange our mechanism that a shall, during the
operation of winding up, become the fulcrum ; and that a force shall act
upwards at b, exactly equal to the downwards pressure which b sustained
during the clock's ordinary motion, the point b being not fixed (as before)
but moveable. The mechanism necessary for this purpose is extremely
simple.

Instead of supposing the pivot b of the barrel to turn in a hole in the
clock-plate, let it turn in a hole in the arm of a frame, fig. 3, of which one
side is bad, and which is itself a lever whose fulcrum projected in a is the
line joining two pins turning in holes of the clock-plate, corresponding

E E2

220 Mr AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE.

exactly in position to the point of the barrel from which the weight W
depends. Suppose another weight w to be suspended from d, of such
magnitude that it will exactly support (acting with the fulcrum a) the
pressure which the lever-frame sustains at b. It will readily be remarked,
that if the lever-frame be bent, as shewn in the figure, no nice adjustment
of the weight of w is necessary. For, if the weight of w be a little too small,
the preponderance of the pressure at b will depress b, and will thereby throw
d so far in the horizontal direction that the increased power of the lever ad
will enable the same weight w to balance the pressure at b. In like
manner, if the weight w be somewhat too large, the approach of d in the
horizontal direction, towards the vertical passing through a, will diminish
its statical momentum, and thereby restore the equilibrium. The effect
of either of these changes on the action of the wheel-teeth is to withdraw
them from the teeth of the pinion by an almost infinitesimal quantity, of
which the effect in practice is wholly insignificant.

We may therefore now be assured that we have provided a force acting
upwards at b, exactly equal and opposite to the pressure which the fulcrum
at b sustains during the ordinary motion of the barrel (inasmuch as our
force does veritably support that pressure during the ordinary motion of
the barrel), and competent to act with insignificant diminution of amount
even if b be moved. One condition therefore of the change of lever-action
is entirely satisfied. The other condition requires that the point a of the
toothed-wheel shall be made, during the process of winding up, the ful-
crum upon which the toothed-wheel turns for the time. But the corre-
sponding point a of the lever-frame is the fulcrum upon which the lever-
frame is able to turn for the time. Consequently all that is necessary to
satisfy this condition is, to contrive that, during the process of winding up,
the toothed-wheel shall be so connected with the lever-frame that it shall
have absolutely the same motion which the lever-frame has on the fulcrum
a; or, in other words, that the toothed-wheel and lever-frame shall (for the
time) move all in one piece. All that is requisite for this purpose is, to
make a ratchet in the inside of the ring of the toothed-wheel ; and to make
a click f to fall in the teeth of the ratchet, its center of motion being some
convenient pointy of the lever-frame. For then, upon taking off the pres-
sure of W and the consequent pressure downwards on b, the pressure of w

Mr AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE. 221

will immediately depress d till the end of/"is firmly lodged upon a tooth
of the ratchet : and then, inasmuch as the toothed-wheel is carried by the
lever-frame at its center b and is thrust by the click f connected with the
lever-frame, the continued descent of d will carry the toothed-wheel in just
the same manner as if it were part of the lever-frame ; and therefore the
toothed-wheel will for the time revolve about a.

The two conditions therefore, which are required in the change of forces
acting on the lever ab c are entirely satisfied ; and therefore the pressure at
c during the winding up of the clock will be the same as during the ordi-
nary going of the clock.

The following description of the movement may perhaps facilitate the
understanding of the action of this mechanism.

While the clock is going in its ordinary way, the weight W descends,
turning the barrel and wheel in such a direction that the teeth of the inter-
nal ratchet glide under the click f without producing on it or sustaining
from it any effect. The action of the weight W and the resistance of the
pinion at c produce a certain pressure on the lever-frame at b which causes
the end d to assume a determinate position, in which it remains without
motion so long as the weight W acts.

As soon as the pressure of W is relieved, the pressure on b ceases ; the
weight w preponderates, the end d drops, the end of f is thrust firmly
against a tooth of the ratchet, and the continued action of w causes the
toothed wheel to turn in a piece with the lever-frame round the center a,
and thereby to produce a pressure at c, which, if a correspond exactly to
the place at which W acted on the barrel, is exactly equal to the pressure
which formerly actd at c.

If the action of W be suspended for a long time, the continued descent
of d will bring d nearer to the vertical passing through a, and will thereby
diminish the statical moment of w, and consequently will diminish the
pressure at c. In this regard the action of this mechanism is exactly similar
to that of the going-spring in Harrison's going-fusee.

222 Mb AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE.

One important point to which I have not yet alluded is the manner of
winding up. It has been supposed all along that the act of winding up
simply relieves the barrel from the pressure of W. This cannot be done
by a square and winch upon the axis b in the usual way. For the action
of the hand in winding up would then produce a force which may be
resolved into a couple acting on the barrel and a force of variable direction
acting at b : which differs entirely from our supposed relief of the pressure
of W. But it can be done easily by inseparably attaching a toothed-wheel
// to the barrel, and mounting a toothed-wheel k with its centre of motion
on the clock-plate, so that the center of k shall be in the same horizontal
line with a, and that the teeth of k may work in those of h: the
winding-up-key being applied to the axis of k. For then the act of turning
It produces no effect on the barrel except a pressure upwards at the very
point where the weight of W produces a pressure downwards (any in-
cidental pressure in the direction of a radius of the barrel, arising from the
slope of the surface of the teeth, evidently having no effect on the angular
motion about a). And therefore, as that pressure upwards must necessarily
be equal in magnitude to the pressure produced by W, it follows that we
may consider the pressure of Was simply relieved in this way of winding
up the clock. The wheel k, it is to be observed, may be of any size or any
number of teeth whatever.

The going-fusee is now complete in its action, so far as regards the use
of a determinate weight W. But by a trifling alteration it will be made
perfect for any weight whatever, without requiring any other change when
the weight Wis changed.

Suppose the lever-frame to be so loaded at cl that the lever-frame when
carrying the barrel and toothed-wheel may be nearly in equilibrium about
a. Then the weight w must be in a constant proportion to W. Now it
will be possible always to arrange the suspension of a single weight by a
line with pulleys attached to the barrel and to cl, so that the tension of the
line acting on cl shall be to the tension of that acting upon the barrel in the
constant proportion which may be assumed.

Consequently the action of that single weight, whatever be its amount,
will then produce two forces such as are proper for the action of this going-

Ma AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE. 223

fusee : and therefore upon any change in that weight there will still be two
forces such as are proper for that action ; and an alteration in the suspended
weight therefore will not require any other alteration in the adjustments of
the mechanism.

If it be required, for instance, that the forces corresponding to Wand
w shall be equal, we must adopt the construction represented in figure 4,
which, for its simplicity, may be considered preferable to any other. If
it be required that the force corresponding to w shall be double that
corresponding to W, we must adopt the construction of figure 5. This
is the construction adopted by the mechanic who (under my general
direction) constructed the clock-work of the Northumberland Telescope.
The wheels h and k are, for clearness, omitted in figures 4 and 5.

The length and inclination of the arm ad will depend upon the hori-
zontal distance between the verticals through a and d : and this horizontal
distance will be found by such a calculation as the following. Suppose, (for
instance) the diameter of the barrel to be half that of the toothed-wheel.

W

The force W acting on the barrel will produce a force — at c, and the

3 W

pressure at b will therefore be — — . This pressure acting on the arm ba

of the lever whose fulcrum is a, is to be balanced by the pressure w acting

3 W W 3

at d: or — — - x ba = w x al. Consequently al = — x -ba. Thus in the
2 * * w 2

W 3

instance of fig. 4, where — — 1, al must = -ba; in the instance of fig. 5,

W 1 3

where — = -, al must = -ba. In determining the inclination to be

w 2 4 °

adopted for the arm ad the mechanic must be guided only by the following
considerations : that if ad be nearly horizontal, the failing of power in the
action of the going-fusee (similar to the weakening of a spring by expan-
sion) will be small, but the angle through which the lever-frame must turn,
in order to correct any small error of adjustment, will be large : whereas,
if a d be greatly inclined to the horizon, the action of the going-fusee fails
rapidly during the suspension of the action of W, but a small error of

224 Mr AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE.

adjustment is corrected by a small motion of the lever-frame. I should
think that an inclination of 40 3 to the horizon would be found convenient.

For fully understanding the action of the mechanism, the following
remarks may be useful.

If the wheel k, fig. 3, be forced a little in the direction opposite to that
of winding up, the clock will go for some time without any descent of W.
For, (using a to denote the point of the barrel where the teeth of k act on
those of h), abc may then be considered as a lever whose fulcrum is c: the
force acting downwards on a, will depress b, and will make several teeth of
the internal ratchet glide under f, and will, at the same time, carry d further
in the horizontal direction ; then if the force on k ceases, the force w, acting
now with increased statical momentum, will thrust f against the teeth of
the ratchet, and will maintain the pressure and motion at c, by the motion
of the whole lever-frame and toothed wheel round a. In this respect, the
action of this mechanism is similar to that of Harrison's going-fusee.

If the distance ba (using a to denote the fulcrum of the lever-frame) be
greater than the radius of the barrel, the force which acts on c during the
winding up, is greater than that which acts during the ordinary going of
the clock. If b a be less than the radius of the barrel, the force which acts
during the winding up is less.

It has been supposed in the whole of this explanation, that b and c are in
the same horizontal line, and that the pressure which the teeth of the wheel
exert on these of the pinion is to be upwards. If the pressure is to be
downwards, the only difference in the form of the construction will be, that
the lever-pivot a will be between the wheel-center and the pinion-center,
and that the inclined arm ad of the lever-frame will be turned towards the
pinion ; its length, &c. will be determined by the same considerations as in
the case which has been fully treated. If b and c are not in the same
horizontal line, all that is necessary is, to make the barrel-line pass over a
pulley, so that the direction of its action shall be perpendicular to the line
be : no alteration whatever is needed for the arm ad, or the line which acts
on it. An instance of such a case is represented in figure 6.

Mr AIRY ON A NEW CONSTRUCTION OF THE GOING-FUSEE. 225

In all cases, the center of the pinion, the center of the toothed-wheel,
the pivot of the lever-frame, and the center of the winding-up-wheel,
must be in the same straight line: and the pivot of the lever-frame,
the place at which the cord is a tangent to the barrel, and the place at
which the teeth of the winding-up-wheel act on those of the barrel-wheel,
must (as projected on the clock-plate) coincide.

I shall terminate this paper by referring to the two isometrical draw-
ings of the new going-fusee, in figures 7 and 8. The first, fig. 7, re-
presents the lever-frame with the barrel, toothed-wheel, internal ratchet,
and clicks, as viewed from the side on which the clock is wound up. The
clock-plate is supposed to be taken off: and, as the winding-up-wheel k
is carried by the clock-plate, the support of that wheel is not represented.
The second, fig. 8, represents the whole of the mechanism, as viewed
from the side opposite to that on which the clock is wound up : the clock-
plate opposite to the winding-up-side being taken off.

G. B. AIRY.

Royal Observatory, Greenwich,
Feb. 5, 1 840.

Vol. VII. Pari II. F F

T"i n.<>>. ■■(tons of the I'ninh /'//// .^or/eJA Vol 1/7 /'/ J

rig. :y

Fig. 4.

/'/,/ />

«T *

/•<>. 8.

Mrfr<Ufc t Pernor, Uthff " CtLtniriUy*

TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

Volume VII. Part III.

CAMBRIDGE:

PRINTED AT THE PITT PRESS;

AND SOLD BY

JOHN WILLIAM PARKER, WEST STRAND, LONDON;

J. & J. J. DEIGHTON; AND T. STEVENSON, CAMBRIDGE.

M.DCCC.XLH.

XV. On Spurious Rainbows. By W. H. Miller, M.A. F.R.S. Professor
of Mineralogy in the University of Cambridge.

[Read March 22, 1841.]

The sixth volume of the Transactions of the Cambridge Philo-
sophical Society contains a Memoir by the Astronomer Royal, on the
Intensity of Light in the neighbourhood of a Caustic, in which the
relative distances of the brightest parts of the first spurious bow, and of
the first and second dark rings, from the geometrical place of the bow,
are determined by calculations founded on the undulatory theory. The
numbers to which he finds these distances proportional are,

Brightest part of bow 1.08

Dark ring between the bow and the first spurious bow 2.48

Brightest part of the first spurious bow 3.47

Dark ring between the first and second spurious bows 4.4 (probably).

It appears also, that the illumination extends beyond the place of
the geometrical bow, the intensity of the light there being about 0.442
of the intensity at the point of maximum brightness.

In order to compare these results with observation, I employed the
method of exhibiting rainbows and the accompanying spurious bows,
invented by M. Babinet (Poggendorff's Annalen, B. xli. S. 139). When
a beam of light admitted horizontally through a narrow vertical slit falls
upon a vertical cylindrical stream of water, portions of the primary and

Vol. VII. Paet III. Gg

278 PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

secondary rainbows and of a large number of the spurious bows may
be seen either with the naked eye or through a telescope, forming a
series of vertical coloured bands, arranged in a horizontal line to the
right and left of the point opposite to that from which the light is
transmitted. A graduated circle placed horizontally with its center in
the axis of the cylindrical stream, carrying a small telescope parallel
to its plane, and having its object-end about one inch distant from the
axis of the circle, served to measure the angle between the line of light
and any one of the luminous bars.

The diameter of the stream was determined in the following manner.
A lens of about 0.77 inch focal length was placed between the object-
glass of the telescope and the stream, at the distance of its focal length
from the axis of the latter, and the angle which the diameter of the
stream subtended when seen through the lens, measured. Next, a scale
of millimetres divided on glass, being placed in the focus of the lens,
the angle subtended by two lines distant one millimetre from each other
was measured. From these two angles the diameter of the stream may
be readily calculated.

In the first observations the diameter of the stream was about 0.022
inch, and the light used was that of the Sun. The mixture of different
colours rendered it very difficult to fix upon the brightest parts of the
bars, especially of those corresponding to the principal bows.

The mean of eight observations of the primary and two of the
secondary gave,

Radius of the brightest part of the primary bow 41.32

Radius of the brightest part of its first spurious bow 40.27

Radius of the brightest part of the secondary bow 51.58

Radius of the brightest part of its first spurious bow 53.57

If 3 (sin 0) 2 = (2 + n) (2 - m), m sin 0' = sin 0, the radius of the geo-
metrical primary bow of the colour corresponding to the index m will be
40'— 20 ; and if 8(sin-v|,) 2 = (3 + /*) (3 — m), m sin ^' = sin \//, the radius
of the geometrical secondary bow of the colour corresponding to the
index n will be tt + 2\// — 6>//.

PROFESSOR MILLER, ON SPURIOUS RAINBOWS. 279

According to Fraunhofer (Denkschriften der K. Akademie der
Wissenschaften zu Miinchen fur die Jahre 1814 und 1815. S. 214, 224.)
the brightest part of the solar spectrum lies between the lines D, E, at
a distance of between one-third and one-fourth of DE from D; and the
indices of refraction of water for the lines D, E, are 1.33358 and 1.33585
respectively. Therefore, for the brightest part of the solar spectrum the
index of refraction of water will be 1.33424. Hence the radii of the
geometrical primary and secondary bows will be 41°.53',9 and 51°.12',9
respectively.

The theoretical distances of the brightest part of a bow and its first
spurious bow from the geometrical bow are as the numbers 1.08 and
3.47. In the primary bow the difference between the radius of the first
spurious bow and the radius of the geometrical bow is 1°.27'. Therefore,
according to theory, the distance of the primary from the geometrical
bow is 27', or the theoretical radius of the brightest part of the primary
is 41°.27'. The observed radius is 41°. 32'. Hence the observed place of
the primary is 5' nearer to the geometrical bow than its place as assigned
by theory. In like manner the theoretical radius of the brightest part
of the secondary bow is found to be 52°.6'. Hence the observed place
of the secondary is 8' nearer to the geometrical bow than its theoretical
place.

In a second series of observations, the eyehole of the telescope was
covered with a red glass which transmitted light from the least refran-
gible end of the spectrum nearly up to the line D. The points selected
for observation were the dark bands and the brightest part of the prin-
cipal bow. The dark bars could be seen very distinctly, and were easily
bisected. Considerable difficulty was, however, still felt in fixing upon
the brightest part of a principal bow, on account of its breadth and the
want of a symmetrical distribution of light on both sides of the brightest
point. An inspection of the results will shew that the latter was sub-
ject to considerable uncertainty. All the observations were liable to be
affected by a sudden shifting of the bars, which was seen occasionally
to take place through a small space to the right or left. The angular

GC2

280

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

distances of as many of the dark bars as could be conveniently observed,
are given in order to meet any future calculation of their places. The
observations of the points at which the bows appeared to commence,
present, as might be expected, great discrepancies ; they shew, however,
that in every case the illumination extends considerably beyond the place
of the geometrical bow. The spectrum seen through the red glass fades
much more abruptly from its brightest point towards the more refrangible
end than towards the less refrangible end. JV being the more refrangible
end, M the brightest point, L that at which it begins to fade, K the
least refrangible end ; the indices of those points deduced from the
double deviations through a hollow prism having an angle of 66°.22' are,
at K 1.3294, at L 1.3310, at M 1.3322, at N 1.3334. The best single
equivalent index will probably be about 1.3318.

(A)

Primary bow, seen through red glass, for which it has been assumed
that n = 1.3318, and .-. 4 ■$' — 2 cf> = 42°.15'. Diameter of the cylinder of
water = 0.0206 inch.

Dark band

1

2

3

4

5

6

7

8

9

10

11

12

13

14

42° .
41 .
41 .
40 .
39 .
38 .
38 .
37 .
37 .
36 .
36 .
36 .
35 .
35 .
35 .
34 .

50'

49
8
16
36
57
25
54
24

59
31

7
45
21

1
40

64'
50
10
17
37
58
27
50
22
56
29
3
37
12

36'
51

8

19
37
58
26
54
28
58
33

6
43

46'
45
2
10
27
50
18
46

53'

58

11

19

37

60

26

61'
54
6
18
33
58

50'
51
4
13
33
53

52'
55
7
15
32

55'

45

8

30

64'
56

7
17

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

281

(B)

Secondary bow, seen through red glass, for which it has been assumed
that « = 1.3318, and .-. tt + 2 ^ - 6 v// = 50°.34'. Diameter of the cylinder
of water = 0.0206 inch.

Brightest ..
Dark band

1

2
3

4
5
6
7
8

49°
51
52
54

55 .

56 .

57 ■

58 .

59 •
59 •

65'

30

36

2

19
23
24
18
10
56

53'

27

39

10

26

30

33

82

54'
16
36
1
16
26

51'
20
37
6
25
30

65'
30
37
8
22

58'

21

40

12

30

68'
29
33

7
25

In a third series of observations, the Sun's light, after being trans-
mitted through a vertical slit 0.25 inch wide, was received upon a prism
distant about 24 feet, having its edge vertical. A second slit also 0.25
inch wide being placed immediately behind the prism, a tolerably pure
spectrum was formed. The stream of water was then placed at the dis-
tance of about 18 feet from the prism, nearly in the brightest part of
the spectrum : and the index of refraction of the rays that fell upon
the stream, determined in the same manner as that of the light trans-
mitted through the red glass.

282

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

(C)

Primary bow. m = 1.3346, .-. 4 0' - 2 = 41.°50',4. Diameter of the
cylinder of water = 0.02105 inch.

Dark band

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

26

27

28

29

30

42°
41
40
40

39
38
38
37
37
37
36
36
35
35
35
34
34
34
33
33
33
32
32
32
32
31
31
31
30
30
30

29'

27
49
4
27
. 51
. 21
. 52
. 25
.
. 34
. 11
. 48
. 22
. 4
. 43
. 23
. 3
• 47
. 28
. 8
. 53
. 36
. 18
. 1
. 34
. 28
. 5
. 53
. 38
. 24

55'
31
53

5
28
53
25
55
29

4
41
17
55
34
12
53
33
IS
55
37
19
58
41
25

6
53
32
18
65

24'
26
50

3
26
52
22
54
27

3
38
14
52
32

9
48
27
10
50
33
13
56
40
23

5
50
29

34'
33
52

4
27
54
25
55
29

4
40
17
55
32
11
52
33
13
52
35
23
58

52'
25
53

7
28
56
24
57
31

5
42
18
57
36
14
55
35
18
58
40
22

43'

25

51

3

25
52
20
52
24

36
11

48
28

8
46
26

8
47

51'

27

52

5

26

22
54

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

(D)
Secondary bow. n = 1,33464, /. ir + 2 ^
of the cylinder of water = 0.02105 inch.

6^' = 51°.19'. Diameter

Limit

Brightest

Dark band 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

50°

13'

7'

30'

57'

19'

8'

47'

51

59

59

57

69

50

49

57

53

5

2

6

6

5

5

6

54

27

23

29

30

28

25

31

55

36

31

38

39

37

32

36

56

35

30

38

38

35

35

36

57

29

25

42

32

30

29

30

58

19

13

23

22

22

20

21

59

6

11

13

10

8

8

59

50

45

56

55

55

51

50

60

33

29

40

38

41

61

15

12

22

17

20

61

52

50

62

58

59

62

32

29

41

36

42

63

9

3

17

17

17

63

43

39

52

49

51

64

18

15

29

25

28

64

52

49

62

61

65

25

24

37

32

65

57

55

71

68

66

29

26

45

66

63

54

76

67

32

26

43

68

68

35

284

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

A fourth series of observations was made with a smaller cylinder of
water, the diameter of which is rather uncertain, the tube having been
accidentally broken before the observations for determining the diameter
of the stream were repeated. At the commencement of the observa-
tions it was found that m = 1.33453 ; at the conclusion it was found
that p = 1.3348. This shews that either the prism or the stream had
been displaced in the interval. The comparison of the observed and
theoretical radii has been made with both values of l x. The former of
these agrees best with the theory.

(E)

Primary bow. If M = 1.33453, 4 <£' - 2 (p = 41°.52'. If M = 1.3348.
4 0' — 2 <p = 41°.49'. Diameter of the cylinder of water = 0.0135 inch.

Limit

Brightest

Dark band 1

2

3

4

5

6

7

8

9

10

11

12

13

14.

15

16

17

18

19

20

21

22

42°

41

40

39

38 ,

37

37

36

35

35

34

34

33

33

32

32

31

31

31

30

30

29

29

29

68'
18
34
28
38
52
12
34
56
23
51
20
49
21
53
26
58
34
9
44
20
56
30

67

' 57'

65'

20

18

19

32

31

33

29

31

28

39

39

38

54

54

54

13

14

12

34

36

35

58

62

60

21

28

26

51

57

...

19

28

50

57

. ..

20

27

25

54

58

57

26

31

30

• •

62

64

..

39

37

..

11

. s

44

••

25

3

77
30
34

29
40
53
13
35
58
25
53
21
50
24
55
32

64'

17
32

29
38
53
12
31
61
29
53
22
52

55

48'

19

34

31

39

53

14

34

61

27

PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

28.5

(F)

Secondary bow. If m = 1.33453, r + S>-6f- 51°.17',5. If
v = 1.3348. tt+2>//-6^' = 51°.23',3. Diameter of the cylinder of water
= 0.0135 inch.

49°

52

53

55 .

57

58

59 •

60 .

61 .

62 .

63 .

64 .

65 .

66 .

53'
26
38
35
4
26
38
44
48
54
48
41
34
25

35'
20
36
28
3
23
37
44
48
47
51
39

63'

38
32
1
21
33
32

44

13'

6

32

30

4

19

32

39
43

44

48'
14
40
30
6
25
42
48
55

48'
18
43
35
9
25
35
43

58'
26
35

29

13
37
39

Dark band

1

2
3
4
5
6
7
8

.9
10
11
12

According to theory, the distances of the brightest part of a prin-
cipal bow and of the 1st and 2nd dark rings from the geometrical bow,
are as 1.08; 2.48; 4.4. Whence, knowing the calculated radius of the
geometrical bow and the observed radius of the first dark ring, the
theoretical radii of the brightest part of the principal bow and that of
the second dark ring may be found. In the following comparison of
these with the mean of the results obtained by observation, it will be
seen that the differences between theory and observation are not greater
than might reasonably be expected. It will, however, be remarked that
in every instance the observed principal bow is a little nearer to the
geometrical bow than theory indicates. This, if not accidental, may be
due to an error in pointing, occasioned by the want of symmetry in
the distribution of the light in the principal bow.
Vol. VII. Part III. He

286 PROFESSOR MILLER, ON SPURIOUS RAINBOWS.

Obs. Theory.

(40'- 20 42 . 15)

rad. primary 41 . 51,4 41 . 45,4

rad. 1st dark ring 41 . 7

rad. 2nd dark ring 40 .16 40 . 14,4

(7T + 2\^ - 6\f,' 50 . 34)

rad. secondary ... 51 . 25 51 . 27,5
(Z>)

rad. 1st dark ring 52 . 37

rad. 2nd dark ring 54 . 7 54 . 12

(40'-20 41 . 50,4)

rad. primary 41 . 27,7 41 . 24,7

rad. 1st dark ring 40 . 51,4

rad. 2nd dark ring 40 .4,4 40 . 5,7

(tt + 2\|/ - 6\|/' 51 . 19,2)

rad. secondary ... 51 . 57 52 . 5,3

rad. 1st dark ring 53 . 5

rad. 2nd dark ring 54 . 27,6 54 . 27

(40'-20 41.52? 41.49?)

rad. primary 41 .20 41 . 18 41 . 15

rad. 1st dark ring 40 . S3

rad. 2nd dark ring 39 . 29 39 . 32 39 . 31

(tt + 2\// - 6^' ... 51 . 17,5? 51 . 23,3?)

rad. secondary ... 52 .16 52 . 18,5 52 . 21

rad. 1st dark ring 53 . 37,4

rad. 2nd dark ring 55 . 31,3 55 . 26 55 . 21

W. H. MILLER.

St. John's College,
Bee. 14, 1840.

XVI. On the Foundation of Algebra, No. II. By Augustus De Morgan,
F.R.A.S., F.C.P.S.; of Trinity College; Professor of Mathematics
in University College, London.

[Read November 29, 1841.]

In presenting to the Society a continuation of the Paper on the
Foundation of Algebra, printed in Vol. vn, p. 173, I wish to make
the principal point of the new communication, which is the filling up
of an unfinished difficulty of the old one, subservient to such a view
of the transition from semi-logical to logical algebra as may perhaps
be useful to any one who may hereafter have to deal with an unexplained
result. By the semi-logical algebra I mean the ordinary science, in
which the explanations are insufficient to include J- 1 ; and in which
therefore the results, though always intelligible when J- 1 disappears,
can only be considered as demonstrated upon the assumption that the
symbolical laws of algebra must in some, though an unknown, manner,
admit of a wider explanation.

The first step to logical algebra is the separation of the rules of the
ordinary science from its principles, or rather of its laws of operation
from the explanation of the symbols operated upon or with. As far
as I can see (and I believe no writer has professed to throw together
in one place every thing that is essential to algebraical process) the laws
of operation are as follow :

1. The literal symbols, a, b, c, &c. have no necessary relation except
this, that whatever any one of them may mean in any one part of a
process, it means the same in every other part of the same process.

HH2

288 PROFESSOR DE MORGAN, ON THE

2. The sign = is the only one of which the explanation is requisite
in the art of operation : it signifies an assertion of identity of operative
effect, and gives the right to substitute one side for the other, when
desired. Its use implies a postulate, the only one demanded : that a = b
gives A = B whenever A is derived from a by the same operations in
the same order, which produce B from b.

3. The signs + and — are opposite in effect ; what one does the
other undoes : and is the symbol of a pair of such opposite operations
having been performed. Thus + a - a = 0. And such operations are
convertible in their order: thus + a — b + c = + c — b + a= —b + c + a, &c.

4. The signs x and +- (or any substitutes for them) are opposite
in effect: and 1 is the symbol of a pair of such opposite operations
having been performed. Thus x a -*- « — 1. And these operations
are also convertible in their order : thus

x a -r- b x c — xc-rixfl= -+ A x c x «, &c.

5. The operations x and -J- are of a distributive character, when
performed upon the results of the operations + and — . Thus

( + a) x ( + i — e) "m ( + a) x ( + b) + ( + a) x ( - c), &c.

6. Like signs ( + and — ) produce + in all cases, and unlike signs
— . And like signs (x and ■*•) produce x in all cases, and unlike signs
■*■ . And each pair of signs is, relatively to its own set, distributive.

7. The signs and 1 may themselves be considered as subjects of
operation, and 1 + 1 is abbreviated into 2, 1 + 1 + 1 into 3, 1 + 1 + 1 + 1
into 4, and so on.

8. The laws by which the symbol a b is used are a b x a c = a h+c and
(a b ) c = a hc .

I believe the preceding rules to be neither insufficient nor redundant,
though I should be noways surprised to see them proved both the one
and the other; least of all if it were the latter.

The most remarkable point in this separation is that the laws of
operation prescribe much less of connexion between the successive symbols
a + b, ab, and a h , than a person who has deduced these laws from an

FOUNDATION OF ALGEBRA. 289

arithmetical explanation would at first think sufficient. The only con-
nexion between the two fundamental operations of + and x is contained
in a(b±c) = ab±ac, and though from this it is a demonstrable identity
that

abbreviated.

a +- a + a + "(1 + 1+1+ •••) x «,

which establishes a connexion between + and x when one of the factors
is derived solely from 1, yet it leaves the general symbol ab, when neither
a nor b is so derived, apparently more free of the meaning of a + b than
any one would predict it ultimately must be: while a h is still less
connected with its predecessor ab. I shall now examine the manner
in which this independence of the three operations has acted in the
explanations which have appeared.

Choosing a unit-line of arbitrary length and direction, and signifying
by A or {a, a), a line of a units in length inclined to the unit-line at
an angle a, it is well known that an explanation can be given, under
which the preceding laws of operation become real consequences of real
conceptions. And it is worth stopping to note that the art of operation,
previously to the explanation of its symbols, is precisely what Dugald
Stewart imagined every mathematical science to be, namely, a pure
consequence of definitions, which upon other definitions might have
been another thing. This opinion was not, and perhaps is not, without
its followers : but I think it will hardly, in any mind, stand the test
of a comparison of any one mathematical science with the purely technical
algebra, which is rigorously founded upon definitions. By itself, this
method of operation, this algebra of rules without meaning, is no more
of a science than the use of the well-known toy called the Chinese
puzzle, in which a prescribed number of forms are given, and a large
number of different arrangements, of which the outlines only are drawn,
are to be produced. Perhaps a dissected map or picture would be a
still better illustration : a person who puts one of these together by
the backs of the pieces, and therefore is guided only by their forms,
and not by their meanings, may be compared to one who makes the
transformations of algebra by the defined laws of operation only : while
one who looks at the fronts, and converts his general knowledge of

290 PROFESSOR DE MORGAN, ON THE

the countries painted on them into one of a more particular kind by
help of the forms of the pieces, more resembles the investigator and the
mathematician.

Mr. Warren, in his explanation* and Dr. Peacock, in his interpre-
tation*, of the algebraical symbols, have both obtained the symbols
a + b and ab independently of each other as to their meaning : while
both, to obtain the meaning of the symbol a b , have had recourse to the
fundamental derivation from a, aa, aaa, &c. The consequence is, that
while both establish most completely the ordinary forms of algebra,
neither is prepared to consider a h where b is other than what answers
to the positive or negative number or fraction of the semi-logical algebra.
Mr. Warren, who carefully avoids all interpreted results, and whose work
is as complete a succession of consequences from explanations adequate
to the results as that of any professedly arithmetical algebraist, has there-
fore totally avoided the use of such a symbol as e e V _1 , using instead

/ \ —

( 1 ) 2,r > a new convention, derived from the roots of unity. Dr. Pea-
cock, making use virtually of the equation

cos + J- 1 sin 9 = (cos 1 + J- J sin l) fl ,

and denoting cos 1 + J — l sin 1 by e, is able fully to interpret all results
arising from e e = cos 9 + J — 1 sin 9, and to prove this equation as a
consequence of the laws of operation. Both writers would consider
e^V-i = cos 6 + J — I sin 9, as an equation resembling — (— A) = A in
ordinary algebra, of which the first side, known to be the same symbol
as the second, can only receive its explanation from the second. And
we see that the complete independence of the explanations or interpre-
tations of a + b and ab leads (in the works alluded to) to a full and
satisfactory account of their properties, while the derivation of a b from
ab ends in a partial and insufficient notion of the meaning of the symbol
itself. There is something disappointing in the first-mentioned circum-
stance, since the mind naturally looks, in the most extensive view of the
subject, for the prototypes of those analogies and modes of derivation
which were of such essential use in the more bounded science : but at the

* I use these words in the same sense as in my last Paper.

FOUNDATION OF ALGEBRA. 291

same time the power of adhering to the modes of derivation of the partial
view is too dearly paid for by a want of generality in the general one.

In my last Paper I pointed out that the analogy of the definitions
of a + b and ab in arithmetic and algebra was perfect, insomuch that,
by an abstraction of the subject-matter of the former from those de-
finitions, the remaining words make definitions which will equally apply
to both views of the science. In fact, a + b is in both, a direction to
do with a what must be done with to make b ; while a b is a direction
to do with a what must be done* with 1 to make b. I now proceed
to disengage a h from its partial dependence on ab, and having established
an independent definition, to examine the analogies which exist between
«* in the ancient and modern view of the subject.

Let R = {r, p), be a line of r units inclined to the unit-line at the
angle p ; and this being r cos p + r sin p J—l, let r cos p = R x , r sin p — R y .
It is in our power to suppose this line given by means of another,
R' = (r',p), by the conditions R x ' = <p(r,p), R v ' = ^(r,p), (p and ^ being
known functions, from which r and p can be determined in terms of
r' and p. The second line may be called the determinant of the first,
and the first line may be said to be determined from the second. Now
supposing only the operation + to have been defined in its most general
sense, we have from every form of <p and >//■ the means of instituting
a new process, as follows. Instead of adding two lines, add their de-
terminants, and let the sum of the determinants be the determinant of
a new line. If (r, p), (s, <r), be the given lines, and (t, t) the determined
line, we have then

(t, t) = <p (r, p) + <p {s, a), x// (t, t) = ^ (r, p) + y}s (s, cr).

* The most analogical view of a b is not a natural one, owing to the idea of a logarithm
being made subsequent to that of an exponent. But if the notion of a logarithm were ob-
tained, prior to the definitions of algebra, from two continuous linear motions, which severally
give equal increments in equal times, the one in difference and the other in ratio, the
exponent obtained from them would first enter as a logarithm, and would always retain
the character of a logarithm rendered into numbers, just as sin -1 A always retains that of
an angle rendered into numbers. Hence, a'° gt or o 1 " 8 ", which are the same things, would
be defined from e, introduced in the first process, as follows : a l0£ b is what arises from doing
to a that which must be done with e to form b.

292 PROFESSOR DE MORGAN, ON THE

If we look for that system in which the newly determined line is
the product, RS, or * (r, p) x («, a) or (rs, p + a), we find that we must
have cp (r, p) = log r, \J/ (r, p) = p. The system of logarithms is meant
to be the purely arithmetical one, and for reasons of numerical con-
venience, the base e is taken, and the angle is measured by the ratio
of the arc to the radius. This species of determinant is what should
be called the logarithm of (r, p) ; but, considering it desirable to retain
the word logarithm for purposes purely numerical, I should prefer to

call (\/(logr) 2 + p 2 , tan - ' p^ — J the logometer of (r, p). All this would

throw no light upon the general meaning of the sign x , but it leads
immediately to that comprehensive definition of R s or (r, p) (* "\ which
Mr. Warren might have adopted, to the introduction of e 9 ^ -1 , without
creating the smallest flaw in his well-secured title to be considered
as having most strictly adhered to explained definitions only : and which
Dr. Peacock might have regarded as the complete interpretation of every
symbolical result in which an exponent occurs that cannot be laid down
on one side or the other of the unit-line. That definition is as follows :
R s means the line of which the logometer is obtained by multiplying
together S and the logometer of R. Thus, OU being the unit-line,
let it be required to lay down OR 0S . Let OL be the logarithm of
OR, and ML the arc of z ROU (rad. OU): then OM is the logometer
of OR. Take OT a fourth proportional to OU, OM, OS, and z TOU
= l SOU + iMOU; then TO is the logometer of the result required.
Place a line of which the logarithm is TV
at an angle whose arc is OV, and that line, ▼

OW, is the one represented by OR os . The
fundamental laws of operation are so readily
established that I do not feel it necessary
to enter upon them ; and the equation
e 9V-! = cos 6 + J — 1 sin 6 is a mere corollary of the definition; for the

logometer of e, or (e, 0) is (1, 0), and (l,0)x^-l or (1, 0) x (fl.^)

* It is to be understood that all operations upon the small letters are those of common
arithmetic.

FOUNDATION OF ALGEBRA. 293

is (o, — 1 , which is the logometer of a line whose length has 0cos-,

or 0, for its logarithm, inclined at the angle 0sin^, or 9. Hence « B ^~'

is a unit of length, inclined at an angle ; or cos 9 + J - 1 sin 0.

It will appear rather against the preceding definition, that it points
out e 6 ^ -1 to signify the same as cos + J— 1 sin 0, whatever e and -w
may be : for there is nothing in the preceding demonstration which
has reference to any particular value of these constants. And, in reality,
as far as this one definition is concerned, and its consequences, there
would be no limitation upon the meanings of w and «?. But when —
having invented this indefinite mode of constructing (1, 6), which leads
us to our result whatever may be e or n, and in fact contains a direct
and inverse use of e which prevents the value of that letter from af-
fecting the result — we equate this mode of producing (1, 0) to a definite
mode derived from another definition, we must expect to see the in-
definite character of the former changed, by the introduction of new
conditions, into the definite character of the latter. But this would
be no answer to the difficulty : for it would be admitting a new
fundamental rule among the laws of operation. Let the reasonableness
of the expectation just alluded to be ground for an assumption, and
we see that •n- (two right angles) and e are connected by the equation
6 V-i = cos 1 + J- 1 sin 1, or e depends upon the angle denoted by 1,
which is all that is necessary. But I say that this equation is a con-
sequence of the whole set of definitions only, without any new assump-
tion. In the first place, it is easily shewn that R s = JR*IlxIl...($ times)
when S=(s, 0) and s is integer. Next, from the definition of multipli-
cation it immediately follows that

cos 80 + J — 1 sin s0 = (cos + J— 1 sin 0) (cos0+ J — I sin 0)...(s times);

whence cos s0 + J — 1 sin s0 = (cos + J — 1 sin 0)';

whence it is cos 1 + J — 1 sin 1, and that only, which, raised to the in-
teger power of s, gives cos s + J — 1 sin * ; for it may readily be shewn
that no other line (r, p) can have the same * th power as (1, 1). It is
then the result of the definitions of addition and multiplication that
nothing but cos 1 + J — 1 sin 1, raised to the power of * (integer), gives
Vol. VII. Pakt III. Ii

294 PROFESSOR DE MORGAN, ON THE

(1, *): it is the consequence of the definition of an exponential operation,
considered apart from the rest, that e^ _1 raised to the power of s, gives
(1, *) : the whole system therefore requires that W -1 = cos ] + J — 1 . sin 1;
which is thus proved previous to the equation of e"^ -1 , and cos 9 + J — l sin 9
from the definition of an exponent. Hence e depends only upon the an-
gular unit, which may be a degree, a minute, a right angle, or any
other, provided that e be taken accordingly. The proof that e = 1 + 1
-s- | + ... , when the angle 1 is that which has an arc equal to the
radius, must be a subsequent matter.

If \ (r, p) represent the logometer, or complete algebraical logarithm,
of (r, p), the equation (r, p) — e x ( r, '' ) is an identical one; for the logometer
of e, or \(e, 0), being (1, 0), say that (t, r) is that of (r, p), whence, by
definition, K l, 0) x (t, r) or (/, t) is the logometer also of t^i r >i > ). Hence,
making 9 — r, we have e T ^ / " 1 = - 1, and taking the obvious truth that
lines equal (both in length and direction) have the same logometers, we
have

M-i)

a proposition which, not many years since, was one of the mysteries
of analysis. It is now a very simple geometrical proposition : the first
side means a line of -k units laid down positively on the unit-line ; the
second side means the logometer of a negative unit turned back through
a right angle. Now the logometer of a negative unit is a line of it units
erected positively perpendicular to the unit-line: whence the identity
of the two sides is manifest.

On the analogy of the complete definition of 72 s with that of a h
in arithmetic, it can only be said that, so far as the latter is intelligible,
it is seen to coincide with the former : while the former itself intro-
duces an element which seems, up to this time, to defy analogy drawn
from arithmetic ; namely, the representation of a projection on the unit-
line by a logarithm, and of one on the perpendicular to it by an angle.
We see how this happens in the deduction of &J-\ = cos 9 + J— 1 sin 9,
and we also see that the general definition of an exponent may be derived
from the idea of exposition (to use an old phrase) of one symbol by an-
other, in such a manner as to reduce multiplication to the result of

FOUNDATION OF ALGEBRA. 295

addition of exponents. The combination of a line, not an exponent,
with one which is an exponent, by the operation newly learnt, or which
might have been learnt, from combining two exponents by the known
operation + , is then obviously natural, and its result completes the
definitions of algebra in their most comprehensive form. But it is
satisfactory to find that the matter is not thus exhausted; and it re-
mains a subject of speculation how it arises that a line perpendicular
to the unit-line has the same relation to an angle which one on the
unit-line has to the logarithm of a length. The following considerations
tend, as far as they go, to give some idea of the origin of this circum-
stance.

Let us go on to the generation of quantities by infinite numbers
of infinitely small elements, premising that nothing will be said which
may not easily be altered into the language of limits by those who
object to the infinitesimal phraseology. Every line (r, p) can only be
formed by addition of equal* elements in one way, namely, that ex-
pressed by f^ {dr, p), p being constant. Let us now consider what takes

place when (r, p) becomes {r + dr, p + dp). This is obviously equivalent

to multiplying the first line by (1 + — , dp), or successively by (1-1 — -, 0)

and (1, dp). The first multiplication alters length only, in the proportion
of r + dr to r\ and answers to multiplying by OK, UK being dr : r.
Now, OU being the unit-line, make UW = dp in
linear units; whence, by the conventions of angular
measurement, OW is (], dp), neglecting the diffe-
rential of the second order by which OW differs from
OU. These ratiunculce (such was the term applied * v v

to differentials so employed when the theory of loga- '
rithms was first explained), UV and UW, being each used in the
multipliers n times in succession, we have resolved the contemporaneous
transitions (linear and angular) into distinct and (if we please) successive
transitions. If we begin with OU, or (1, 0), and proceed through n
multiplications, and make dr : r always the same, and = /n, we have

* Remember that 'equal' means 'same in length and direction.'

112

296 PROFESSOIl DE MORGAN, ON THE

( { 1 + ,.}", ndp) = (1 +m)" • (1 + dp J- 1)" (A).

Now let n be infinite, and let (1 + n)" = r, ndp = p, from which we find

= f, (l + dpj-l)" = e<>^,
M ftp

all following from the assigned laws of operation. Hence re p ^~ x is the
representative of a line r inclined at an angle p ; while log r and p, each
in its own way, and to a different radix, may be considered as a register
of the number of transitions by which we pass from (1,0) to (r, p).
The term logarithm itself, as is well known, is a consequence of a similar
notion of comparison of numbers by the registration of the 'numbers
of the ratios' by which we pass from unity to those numbers. The ratiun-
cula? u and dp must be in the proportion of log r and p, and the line
formed by adding WU and UV, or n + dpj—l, is one which gives these
two ratiuncula? for its two projections. This line repeated n times gives
log r + 9J—1, the logometer of (r, 9) as I have called it. Should any
objection be taken to that term, perhaps the words compound logarithm
might be preferable. Observe, that this derivation of the logometer is
independent of the second side of {A), and might be introduced pre-
viously to the expression of (r, 9) in the form re'v'- 1 .

I have throughout avoided considering the ambiguous values of sym-
bols, a thing for which there is no necessity, as has been frequently
shewn, and as I noted in my last Paper. The more I think on this
subject, the better satisfied do I feel, that the new algebra should have
no symbols of double or multiple value whatsoever; that is, that the
meaning of each elementary symbol should not be considered as com-
plete, unless it expresses the amount of revolution from the unit line
by which it is to be made to attain its direction, as well as that direction
itself. Undoubtedly, after a time, the student should be shewn how to
drop this part of the definition ; but this he will better be able to do
than to take it up after a previous training, which has never intro-
duced it. This is an important point to those who believe as I do, that
it will not be long before the new algebra is introduced into elementary
instruction ; and it is the more important, because there are some new
species of ambiguities altogether peculiar to the most general view, and
which must remain such until further inquiry points out the mode of

FOUNDATION OF ALGEBRA. 297

dealing with them. These last can hardly receive due attention, unless
they are carefully distinguished from the previous and well-known cases
of the same kind ; which will be only done by adopting that system
of definitions which destroys the latter altogether.

ADDITION.

A theorem of M. Cauchy, which is well known to the readers of
Liouville's Journal, by the comparatively easy demonstration which
MM. Sturm and Liouville have there given of it, may be set in so
clear a point of view by the complete algebra, that I here add a de-
monstration of it. This theorem belongs essentially to the complete
system of algebra, as will be evident from its enunciation.

As before, Z = {%, £) or ««W- 1 , means that Z is a line of the length
» inclined at an angle £ to the unit-line.

Theorem. Let <pZ be any function of Z, and let Z = x + y J- 1
give <pZ = p +q J—\. Also, within the whole of
the figure ABCD, its contour included, let <pZ, (p'Z,
&c. never become infinite, x and y being the co-ordi-
nates of any point within it. Let any point be called
a radical point which makes cf>Z or <f>(x + y J — 1) = 0.
In carrying a point in the positive direction of revo-
lution round the contour ABCD, let the fraction " change sign by
/tossing through zero k times from + to - , and / times from — to + :

but let it never change sign by passing through -, that is, let there

be no radical point on the contour itself, and neglect altogether the
cases in which it changes sign by passing through oo . Then the number
of radical points contained within the contour is ^ (k — I).

298 PROFESSOR DE MORGAN, ON THE

Encircle any point P by an infinitely small contour, on which let
a point be carried round P. Four cases arise ; neither p nor q vanishes
within or on this contour ; p vanishes but not q ; q vanishes but not
p : or both vanish.

If neither p nor q vanish, there is never change of sign in either
(for by hypothesis they do not become infinite), and the theorem is
true for this infinitely small contour : for k and / are both = 0, and
there is no radical point.

If p alone vanish, the curve p = (p being a function of x and y,)

passes through the small contour at a single or multiple point : and —

may change sign at those points of the contour through which the curve
passes ; but the fraction always becomes and never oo . There are then
as many changes of sign from + to — as from — to + , and the theorem
is true : for k = I, and there is no radical point.

If q alone vanish, the curve q = passes through the point : and

every thing is as in the last, except that - always becomes oo when it

changes sign. Hence the theorem is true ; for k and / are each = 0, and
there is no radical point.

Lastly, let there be a radical point within, but not on, the contour :
which it is evident may be supposed to contain only one radical point.
Let x — ft, y = v at the radical point, and let Z be the radius vector
drawn from the origin to a point in the contour, and R that drawn from
the radical point to the same point of the contour. If then

m + vj - 1 = A = {a, a);

we have, using the extended system of algebra,

Z = A + R, or (»,£) = («, a) + (r, P ), or ase^V- 1 = ae a ^~ l + ri»v^,

r being infinitely small. Now let <p(A + R), or <pZ, be capable of being
expanded into the series

B R m + B,R m+1 + B 2 R m+ *+

in which, on account of the value of R, we need only consider the first
term B a R m .

FOUNDATION OF ALGEBRA. 299

By our hypothesis m is an integer, and there are m radical points in
one, answering to m equal roots of <pZ=0. Also, B being (b , fi ),
we have

B B m = b^r™ {cos (mp + /3 ) + sin (m P + /3 ) . J- 1 } .

whence * = cot (mp + /3 ).

Now while p goes through a whole revolution, mp + fi passes from
(i to 2mir + /3 through »« complete revolutions, and changes sign 2m
times from + to - passing through each time : but it never changes
from — to + except by passing through oo . Hence k = 2m, 1=0,
l (k - I) = m, which verifies the theorem, since there are m roots within
the contour.

Next, let the whole figure ABCD be divided into an infinite number
of infinitely small figures, with no other limitation than that no radical
point is to fall upon one of the lines of division: and let a point move
round each of the infinitely small figures in the positive direction of
revolution. It is clear that the expression ^ (2A - 2/) will not be altered
if we remove all the internal division lines and leave only the external
contour ABCD : for each internal line is described by two points moving
in opposite directions, and wherever one point adds a unit to 2&, the
other adds one to 2£ Hence the value of Ik - 2/ can be found by
finding that of k — I for the external contour only.

If we suppose cpZ to be rational and integral, say = A Z n + AiZ"- J + ...,
and if we make the contour in question a circle with the origin as a center,
and a radius so great that the highest term A Z" need be the only one

retained, we find that - = cot (ȣ + a ), which gives, as before, in a

revolution, k = 2n, I - 0; whence the whole number of roots of (pZ=0
is neither more nor less than n.

It is easily deduced from the preceding that the number of real
roots of <px = lying between x = a (the less) and x = b the greater, is
the number of vanishing changes of sign from — to + which take place
while x passes from a to b in the quotient of

>*- 0".r£ 4 c^x-V--- ... divided by <j>'x. - <p'"x^- + $>x

2 T 2.3.4 J -r • T 2.3 r 2.3.4.5

300 PROFESSOR DE MORGAN, ON THE FOUNDATION OF ALGEBRA.

y being an infinitely small positive constant : provided that neither a nor b
is a root. This residt, however, would be of little general use: in fact,
this theorem of Cauchy requires an examination of the contour which
is equivalent in that which must be made of the axis to find the real
roots. It makes the examination of a line equivalent to that of the
whole included space ; but does not profess to help in that examination.
But in the important case in which the contour is a rectangle with sides
parallel to the axes, or when it is desired to find all the roots of the form
x + y y/ — 1 in which x lies between given limits, and y between other
given limits, this theorem is a complete reduction of the question to
that of finding the number of real roots of four equations which lie
between given limits, one pair for each equation. It thus supplies the
theoretical desideratum which Fourier and Sturm have left.

A. DE MORGAN.

University College, London,
October 12, 1841.

XVII. An Enquiry into the Causes which led to the Fatal Accident on
the Brighton Railway (Oct. 2, 1841), in which is developed A
Principle of Motion of the greatest importance in guarding agtiinst
the Disastrous Effects of Collision under whatever circumstances
it may occur. By the Rev. J. Power, M.A., Fellow and Tutor
of Trinity Hall, Cambridge.

[Read November 29, 1841.]

When accidents have occurred on railways, in the majority of instances
some cause has been immediately apparent, to which the occurrence might
be reasonably imputed ; but the fatal accident which took place on the
Brighton Railway, in October last, the cause of which still lies buried
in the greatest obscurity, forms a remarkable exception to the above rule.

In meditating on the possible causes of this accident, I have arrived
at some Dynamical Results, which will be given in the course of this
investigation, and which are, as I conceive, of the greatest importance
to the public safety, inasmuch as, by attention to them on the part of
Engineers, the disastrous consequences of collision may be very materially
diminished.

In order to pursue the proposed inquiry with effect, it was necessary in
the first place to ascertain, as accurately as circumstances would permit, the
true history of the accident. With this view, I have examined with great
care the report of the Coroner's Inquest as given in the Times news-
paper, and have succeeded, to my own satisfaction, in forming a tolerably
consistent narrative out of the disjointed materials of which the evidence
is composed.

It would be tedious to occupy the attention of the reader with the
critical details of the above examination, and I shall therefore proceed
Vol. VII. Paet III. Kk

302 Mr POWER, ON THE PREVENTION OF THE

at once to lay before him the historical conclusions at which I have
arrived, so far as they are connected with the object I have in view.

I will take up the narrative at that point where the train, with
two engines in front, was making its way from the Horley Station in
the direction of Brighton, and was proceeding with apparent safety at
the rate of about 30 miles an hour. The leading, or Pilot engine as it is
termed, was a small four-wheeled engine which had been put on at the
Horley Station, in order to afford a temporary assistance to the large
six-wheeled engine which had brought the train from London. I shall
call these the first and second engine respectively, as they occurred
in the order of the train, though it may be worth mentioning, that,
in the evidence of Goldsmith, the driver of the pilot engine, these
terms are used in the inverted sense, the large original engine being
named the first, and the smaller engine, which was subsequently pre-
fixed, being named the second.

On approaching a certain cutting, called " the Copyhold Cutting,"
and at a distance from it of about half a mile, the driver of the second
or large engine turned off his steam, his motive for so doing being,
that he might reserve his steam for the latter part of the way to
Brighton, when he should be deprived of the assistance of the other
engine. By this operation the speed was reduced from 30 to about
20 or 25 miles an hour, and the train was continuing to proceed with-
out any inconvenience over the half mile which intervened between
the place where the steam of the second engine was turned off and
the entrance of the cutting.

Before entering the cutting, a labourer on the road was observed
to hold up his hand, the usual sign for slackening speed. Upon this,
the driver of the first engine turned off his steam " to within half an
inch," an interval which (as the regulator in closing the aperture,
through which the steam enters into the cylinder, moves over a quan-
drantal arc of seven inches radius ',) corresponds to a deficiency of about
^ from the whole extent of the range.

* See description of Stephenson's Locomotive Engine in the new edition of Tredgold on
the Steam Engine.

DISASTROUS EFFECTS OF COLISSION ON RAILWAYS. 303

Immediately after this, the driver of the first engine perceived a
different motion in his engine, which, to use his own expression,
" wavered backwards and forwards," and in a very short interval of
time, the fore wheels were thrown off the rail, causing the engine to
upset, and giving rise to the accident, with all its frightful concomitants.
The motion which preceded the upset of the engine, is described by
the driver of the second engine as a rocking motion, and by a labourer
who viewed it from the road, as a jumping up and down of the fore
wheels. The short period of time for which it lasted was estimated by
the driver of the second engine, and the person who viewed it from
the road, at half a minute ; and though the driver of the first engine
stated that it " lasted only an instant," some allowance must here be
made for the vagueness of language, and we shall probably be not far
from the truth, if we regard it as an interval of very small but sen-
sible duration. It appeared also, by the evidence of one of the engine
drivers, that at the time the accident happened, " the pilot engine was
doing very little work, merely keeping tight the chain ;" from which
incidental expression it may be inferred with certainty, that the two
engines were connected by a chain, which was at liberty to be loose
or tight according to the distance between the two engines*, a fact,
be it observed, which is of the greatest importance in the explanation
which is about to be offered.

It further appeared in evidence, that the engine had been examined
and found to be in perfect order ; and though the driver of the second
engine gave it as his opinion, that the accident was occasioned by clay,
or some greasy substance, lying on the rail, yet it appears by the evidence
of the person whose duty it was to inspect this portion of road, that the
rail was perfectly clean and had nothing slippery upon it, that it was,
moreover, on a bed of sand, and not on clay. The preceding rains had
indeed rendered it a prudent precaution that the train should come steadily
over this portion of road, and such was alleged to be the meaning of the
signal which the man made by holding up his hand, a practice which

* I have lately seen a chain of this description connecting two engines drawing a heavy
train on the Birmingham railway, allowing a considerable variation of distance (to the ex-
tent perhaps of about a foot) between the engines.

KK2

304 Mr POWER, ON THE PREVENTION OF THE

had been adopted for some days previous. It appears also that the rate
at which the train was proceeding was nothing more than the usual
rate, which was well warranted by the sound state of the rail. Indeed,
no danger, or cause of danger, appears to have been suspected by any
person belonging to the establishment up to the moment the accident
commenced.

The first engine was, indeed, described as top-heavy, having been
recently filled with water, but this was nothing more than must usually
occur under similar circumstances, that is to say, when a pilot engine is
first attached.

We have, then, before us a train of carriages (including the large
six-wheeled engine, whose steam had previously been turned off), drawn
at a comparatively moderate rate by a four-wheeled engine in front,
and in the apparent absence of predisposing causes, we have to account
for the continued jumping motion, which was observed to take place
in the front engine upon the driver's turning off his steam.

Now it occurred to me, that the motion described was exactly such
as would have resulted from a series of jolts or moderate impulses com-
municated at the back of the engine, provided they were communicated
at a point considerably lower than its centre of gravity ; and I pro-
pose to show, in the first place, how an impulse from behind com-
municated lower than the centre of gravity would cause the front engine
to lift up its fore wheels ; and secondly, to point out how, under the
circumstances of the case, a succession of such ascents of the fore wheels
might possibly have taken place. For, be it observed, the phenomenon to
be explained is not so much the final overthrow of the engine, as the pre-
ceding jumping motion, or series of jumps, which led to it; a single
jump might, no doubt, have been sufficient to cause the accident, but the
probability of danger arising from a single jump, would be incomparably
smaller than that which would arise from any cause which rendered
possible a continued series of them, such as was in fact observed to take
place on the present occasion.

I shall demonstrate hereafter the following important proposition,
namely, that there exists at the back of a locomotive carriage of any

DISASTROUS EEFECTS OF COLLISION ON RAILWAYS. 305

kind, a point, at which if a horizontal impulse be communicated, no
jumping motion of the kind we are considering can possibly result,
but if the impulse be communicated, at a point lower than this, the
carriage will lift up its fore wheels, and if higher than this, it will lift up
its hind wheels.

Professor Willis has been so kind as to fit up, at my request, a small
model, which affords a most satisfactory experimental proof of the truth
of this proposition*, which I previously arrived at by mathematical
considerations.

If we neglect the mass of the pair of wheels and axle about which
the rotation is supposed to take place, a simple mathematical calcu-
lation suffices to show, that the point of quiescence, or that at which
the horizontal impulse must be applied, in order that no rotatory motion
of the kind we are considering may be impressed, will be situated in the
same horizontal level with the centre of gravity of the carriage. But
since in locomotive engines the wheels and axles are of considerable mass,
(in Stephenson's locomotive the driving pair of wheels and the axle
connecting them weigh about a ton and a quarter,) it was desirable
to ascertain how far this circumstance would cause the point of quiescence
to deviate from the level of the centre of gravity.

The problem then becomes more complicated, but the result of the
investigation shows, that when the rail is regarded as perfectly smooth,
the deviation of the point of quiescence from the centre of gravity of
the whole carriage, including the wheels and axle, is nothing, whatever be
the mass of the latter. The roughness of the rail may however cause it to
deviate slightly below this level, but even in the case of perfect roughness,
the deviation is so slight that in practice it may be safely neglected.

The mathematical details connected with this part of the subject
will be given in the sequel ; and I now proceed to mention a second
principle, which is essential to the solution I am about to offer, and
which, in the absence of sufficient data, I assume rather as a very
probable hypothesis, than as a mathematically demonstrated theorem
like the former. The principle I assume is this, that if the two

* The experiment was exhibited at the time the communication was read.

306 Mb POWER, ON THE PREVENTION OF THE

engines, both having their steam turned off, were started with the same
velocity, the lighter would be retarded more rapidly than the heavier.

If indeed we assume that the machinery is similar in both, the prin-
ciple will be readily allowed ; for the rapidity of the retardation will be
as the retarding causes directly, and as the mass of the engine inversely.

The principal retarding causes when the steam is turned off are,
1st, the resistance of the air ; 2dly, the friction of the parts of the
machinery sliding over one another, the principal of which is, probably,
the friction of the pistons within the cylinders ; 3dly, the roughness
of the rail, that is to say, such retarding causes as may act in the man-
ner of minute obstacles lying upon it ; 4thly, the friction on the axes.

Now, though the effect of the two last causes will be nearly as
the masses of the engines, the two former may be regarded as pretty
nearly the same, the machinery of the engines being supposed similar
in the two cases ; on the whole, therefore, the retarding causes in the
smaller engine will be diminished in a less ratio than its mass ; it will
consequently be retarded more rapidly than the other.

Let us now apply the principles which have been laid down to the
case before us.

It will be recollected that the engines were connected by a chain,
which admitted of being tight or loose as the distance between them
was greater or less: it will be recollected also, that the pilot engine
was described as "top-heavy", which makes it highly probable that
the frame, to which the chain is usually applied at the middle, and
the buffers, (or disks which are brought in contact when the engines
approach each other), at the two sides, was considerably lower than
the centre of gravity of the engine. Lastly, it will be recollected,
that the steam of the large engine had been previously turned off.
Let us now consider what will take place when the steam of the small
engine in front is turned off.

The small engine being retarded more rapidly than the large one
following it, the engines will be brought nearer and nearer to each
other, the connecting chain in the mean time becoming slacker and
slacker, and by the time the buffers are brought in contact, a finite

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 307

difference of velocities will be generated: this will occasion the first jolt,
which being applied at the back of the pilot engine considerably lower
than its centre of gravity, will cause it to lift up its fore wheels. By
the elasticity of the buffers, (which, when attached to engines, are not
usually furnished with springs like those attached to the carriages), the
velocity of approach will be immediately converted into a velocity
of separation, according to the laws of elasticity and impact, and,
for the moment, the front engine will be again driven ahead of the
other; but by the continued excess of the retardation of the first
engine, the velocity of separation will at length be reconverted into a
velocity of approach, giving rise to a second jolt, and occasioning the
front engine a second time to lift up its fore wheels; and the same
process might be repeated a great number of times in succession.

In the above explanation, for the sake of simplicity, I have omitted
the consideration of the effect of the train following the second engine ;
but since the carriages are exempt from one great cause of retardation
to which the engines are subject, namely, the friction of internal
machinery, it is clear that they would, if left to themselves, be retarded
less rapidly than the engines, whence it is easy to see, that the effect
of the train of carriages, will tend to push on the second engine, and in-
crease the effect which has been described.

I have supposed, moreover, that the engine has time to resume its
natural position after each jolt before the succeeding jolt is commu-
nicated, but if the jolts succeed one another more rapidly than the
natural time of an ascent and descent of the front wheels, it is easy
to see how, under favourable circumstances, they might conjoin their
effects so as to increase the angle of elevation very considerably. The
most favourable case for producing this effect, would be, when the
second jolt is communicated at the precise moment when the front
wheels have attained their highest elevation due to the motion im-
pressed by the first, and so on for the third and following jolts.

These successive ascents of the fore wheels were exactly the phe-
nomena which it was our object to account for, being such as were
observed to precede the overthrow of the engine, and such as no doubt

308 Mr POWER, ON THE PREVENTION OP THE

will be attended with the greatest danger whenever they occur. In-
deed it is manifest, that if during the ascent of the fore wheel, any
accidental cause should give the engine at the same time a hitch in
a horizontal direction, the fore wheels must be thrown off the rail,
as in fact took place in the present instance, occasioning the over-
throw of the engine and the disastrous consequences which ensued.

Whether the above be the true solution of the accident it is possible
that a variety of opinions may exist ; but I conceive, no one will doubt
the importance of the principles which have been developed in the
course of this investigation, and which give rise to the following practical
conclusions : —

1. In the construction both of the engines and carriages care should
be taken that the centre of gravity of each engine or carriage be about as
low as the horizontal frame to which the buffers and links are attached.

2. If conformity with the above rule be attended with practical
inconvenience, the same object might be attained by placing the buffers
at the proper height, by means of strong additional frame-work, connected
with and rising from the general horizontal frame.

I am not sure that a single pair of opposing buffers placed mid-way
between the rails, would not be better than two pairs of buffers placed
one immediately over each rail, in order to avoid any tendency to rotatory
motion in a horizontal plane, which any inequality of action in the latter
might occasion. But there may be practical objections to this arrange-
ment with which I am unacquainted.

3. A further means of diminishing the danger would be to shorten
as much as possible the connecting chains, in order to constrain the
engines and carriages to move with the same velocity, and to prevent
the accumulation of any finite difference of velocities between any two
engines or carriages throughout the train*.

* I have lately observed on the Birmingham railway that the buffers of the contiguous
carriages are forced into immediate contact, and the connecting chains made as tight as pos-
sible by means of a screw-power. Why might not the same mode of connection be adopted
when a pilot engine is attached, instead of a loose chain, which appears to be the usual
practice? If injury to the machinery be feared, arising from the jarring vibrations which
would accompany this contact, these might be prevented by furnishing the buffers of the
engines with springs similar to those of the carriages.

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS.

309

G

E

A

c

K

B

E

I

D

p

A

Q

SB

I will now give the mathematical details to which allusion has been
made.

It will be as well to begin with the very simple case of a body
whose centre of gravity is G, sup-
ported on a perfectly smooth hori-
zontal plane PQ by two props BD,
CE, and urged by a horizontal
impulsive force F at the point A.
Draw GHKL vertical.

Let GH = h,
DL = a,
GL = b,

M the mass,

k its radius of gyration about G,

V the horizontal velocity communicated to D,

a the angular velocity about D resulting from the impact.

Since the velocities communicated to G are
aa vertically upwards, and
V ' — ba horizontally forwards;
we have by the usual principles of motion,

M(F-ba) = F,

Maa = R,
Mtfa = Fh - Ra,
R denoting the vertical reaction at D.

Hence M(a 2 + k*)a = Fh,

Fh
a M (a 2 + k 2 ) '
._ F . F a* + k 2 + bh
r= M + ba= M a~V¥

Consequently, if h is nothing, no angular motion will result from
the shock; this is the case when the horizontal impulse F is directed
through the centre of gravity.

Vol. VII. Part III. L l

310

Mk power, on the prevention of the

But if the direction of F passes lower than the centre of gravity, an an-
gular velocity about D must necessarily result, the magnitude of which
is proportionate to the distance h at which the impulse passes below G.

If the impulse passes above G, h changing its sign, a assumes a
negative value, which is impossible
so long as D is supposed to remain
in contact with the plane; we must
in this case regard E as the point
which remains in contact with the
plane, and calling R' the vertical
reaction at E,

a the distance EL,
a the angular velocity about E,
we have, as before,

Ma a' - R\
Mk*a' = Fh - R'a';
whence M (a' 2 + ¥) a' = Fh,

a Mia'" + *»)'
which shows, as before, that if the direction of F passes through G,
no angular rotation will be communicated; and further, that if the
direction of F passes higher than G, an angular velocity, whose mag-
nitude is proportionate to the distance GH, will be generated, by
virtue of which D will be carried upwards, E remaining in contact
with the plane.

The preceding results may be regarded as near approximations to
the truth, in the case of a wheeled carriage, when the mass of the
wheels is inconsiderable compared with the mass of the carriage.

But as the wheels and connecting axles in locomotive engines are
very massive, it may be useful to inquire what influence the mass of
the wheels and axle, about which the rotatory motion takes place, may
have in modifying the preceding results.

The carriage with its wheels not constituting a single rigid body,
as in the last case, the problem becomes much more complicated, and
it is extremely difficult to avoid sources of error in applying to it the

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 311

same formulae for the motion of rigid bodies, as were used in the
preceding example. On this account I prefer treating it by the method
of Lagrange.

Let us suppose the whole mass of the engine to be projected upon
a vertical plane parallel to the direction of the rails, and that a hori-
zontal impulse F from behind causes it to lift up its front wheels
and rotate about its hind or driving pair of wheels.

Let x, y be the horizontal and vertical co-ordinates of any point
m of the carriage, deprived of its hind pair of wheels and their con-
necting axle, referred to a fixed origin behind the carriage.

x\ y those of any point m of the hind wheels and axle ;
u, v the horizontal and vertical velocities communicated to m by
the impact;

u', v' the same for m'.

By D'Alembert's Principle, the momenta subject to the conditions
of equilibrium are

F, — mu, — &c. — m'u', — &c. horizontal;

and — mv, — &c. — m'v', — &c. vertical.

Hence, naming x the horizontal co-ordinate of the point of appli-
cation of F, we have by the principle of virtual velocities,

Fix - Z(muSx) - 2(m'u'§x') )

(1),

I'v'ly')) '

Let V be the linear velocity communicated to the axis of the hind
wheels.

a the angular velocity communicated to the carriage about the axis
of the hind wheels, tending to diminish x and increase y.

a the angular velocity communicated to the hind wheels about their
axis, tending to increase x and diminish y.

Id, l& any small virtual angles of rotation of the carriage and hind
wheels in direction of the angular velocities a, a respectively.

8s the corresponding horizontal space described by the axle of the
hind wheels.

L L2

312

Mr POWER, ON THE PREVENTION OF THE

Then if the rail be perfectly smooth the variations 89, 89 1 , 8 s are
independent of each other. But if the rail be perfectly rough, so
that the wheel remains in perfect rolling contact with it during the
shock, 89" and 8s are connected by the equation 8s = r89', r being
the radius of the wheel.

Again, assuming the moveable projection of the axis as a new
origin, let

f , tj be the co-ordinates of m ;

£', »/ be the co-ordinates of ml.
We have, manifestly,

U mm V — r\a,
V = £ct,

tf . V + n'a,
V' = - fa'.

Also, 8x = 8s — ij89,

h - m

M = 8s + v '89',

8y' = - f 56T.

Again, if a, b be the horizontal and vertical co-ordinates of the
centre of gravity of the carriage exclusive of the hind wheels, and h
the vertical distance below this centre at which F passes, we have

8x = 8s - (b - h) 89.

Substituting these values, the equation (1) becomes

= F {8s - {b - h) 89] - 2 {m {V - n*) (^ - r,89 )}

- 2 {m? a89}

- 2{m'(r+ r{d) {Is + r,'89')\

- 2 {m'?*a'89'\

Hence if the rail be perfectly smooth, equating to zero the co-
efficients of the arbitrary variations 8s, 89, 8&, we obtain

(2).

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 313

= F - VZm + aZmr, - V2m' - a'Lm'r,', ~\

= - F.(b- h) + r2ra»/-«2 \m(? + v°)},> (3).

m - VZm'r,' - u'Zm'W* + ?'•). )

If M = 2m = mass of the carriage without the hind wheels and axle ;
M' = 2»?' = mass of the hind wheels and axle;
k the radius of gyration of M about its own centre of gravity ;
&' that of M' about its axle.

Since 2 m r\ = Mb,
•2m'ri= 0,

S»(f + >/ 2 ) - Mia 1 + ¥ + k*) t
2z»'(p + v*) = M'k'*,
the equations (3) above reduce themselves to
= F - {M + M') V + Mba,

= - F(b -h) + MbV- M{a % + ¥ + A 8 ) a,

0= - clM'k'K

The last gives a = 0, which shows, as we might have expected,
that the rail, being perfectly smooth, has no power of impressing a
finite angular velocity by a horizontal reaction.

The two former give by the elimination of V,

h M ' h

F M + W

a =

M „ Mb*

a % + ¥ +

M+M'

If the rail be perfectly rough, the equation (2) becomes by the
above reductions,

0= {F - (M + M') F + Mba] $s

+ \Fh - Fb + MbV - Mia 9 + 6 2 + k*) a] $6

- a'M'k'*W,

314 Mr POWER, ON THE PREVENTION OF THE

with which must be combined the conditions,

r

, V

a = — •
r

Substituting these values of §& and a, and equating to zero the
coefficients of Ss, $0, we obtain

= F-(M+M')V + Mba - M'V.^,

= Fh - Fb + MbV - M{a* + V + #) «.
The former of which gives

F+Mba

V =

M+M'(l+^)

Substituting this in the second, and putting

q = ; re- > we find

E. h ~ g

a " M ' a 3 + If + qb '
Hence in general,

E- LzS

a ~ Ma* + h* + qb'

M'

where q = when -^ = o,

sv when the rail is smooth,

M + M'

M'(l + ^)b

— when the rail is perfectly rough.

M + M (l + £)

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 315

Since -= ™ is the vertical distance of the centre of gravity of

M + M below that of M ; it follows that in the second case, as in
the first, the impulse must pass through the centre of gravity of the
whole carriage M + M', in order that no rotatory motion may be
impressed.

In the third case, if, for greater practical convenience, we wish to
determine the distance (q t suppose) of the point of quiescence below
the centre of gravity of the whole carriage, denoting by ft, the height
of the centre of gravity of the whole carriage above the level of the
axle, we have

Mb Mb

ft =ft -

M+ M'~ M+ M
k'

±d ' I 1 + v) °

M

q ~ q M+ M'~ ~ TTT FT M+ M

Reducing and substituting for b its value in ft,, we find

b ' M' m

v l£_~ M + M'' r*

r 2

If, with Tredgold, we suppose M' = l£

M '' 1? * b ' M' h'* . .

°, vt = ng-nar, • ^ b < ver y nearl y-

M + M + M '

r

r ' = 12/

, tons,

and M + M'

M 1
and at a rough estimate take - = ^, we find q = .006^ Very nearly.

Thus, if b,= 12 inches, q,= .072 inches;

if b, = 18 inches, q t = .108;

if ft, - 2 feet, q, = .144.

Hence we see, that the weight of the wheels and axles, and the
roughness or smoothness of the rail, make no difference perceptible in
practice ; and that in order to ensure the absence of rotatory motion in a

m -w = - Pa '

SI 6 Mr POWER, ON THE PREVENTION OF THE

vertical plane, arising from a horizontal collision, it is necessary and
sufficient that the centre of the buffers should be placed as nearly
as may be on the same level with the centre of gravity of the
engine or carriage.

If we suppose the rotatory motion impressed to be very small, it
is easy to calculate approximately the ascent of the elevated wheel.

M'

Neglecting -^ and calling 9 the small angle of elevation at the

time t, we shall have approximately,

. d'.iaO) P

Li L — jo" 4-

df " *- 3f*
P denoting the vertical pressure at the rail,

__ d*9 ag

Hence -77, = — , ,* ,

dt 2 a 2 + k'

d9 agt

-j- = Const. j-^-yj

dt a 2 + k

Fh - Magt

M{a % + k 2 ) '

the initial angular velocity being „, 2 7^ .

Integrating again, so that 9 is when t = 0, we obtain

2Fht- Magt"

" 2 M(a 2 + k 2 ) '

Hence, 9 attains its maximum value when Fh — Magt = 0, or
t = ^rr — . and at that moment the value of 9 is

Mag' — 2M 2 ag(a' + k 2 )'

If we suppose the centre of gravity to be situated half way be-
tween the two axles, multiplying by 2 a, we find for the extreme
elevation of the wheels, when the disturbance is small, the expression

F*h*

M 2 g(a 2 +k 2 ) ;

DISASTROUS EFFECTS OF COLLISION ON RAILWAYS. 317

a quantity, which it is desirable to render as small as possible, in order
to ensure safety to the engine or carriage under ordinary circumstances.

It has lately been the subject of discussion, whether, by increasing
the distance between the bearings, the proportionate increase of the
linear ascent of the wheel, due to a given angular rotation, would not
increase the danger of running off the rail.

The preceding result shows, that no such danger is to be feared,
but that, on the contrary, the increased linear ascent of the wheel,
due to the greater length of revolving radius, is far more than com-
pensated by the diminution of the angular velocity itself; and, as
regards the comparative safety of four-wheeled and six-wheeled engines,
it shows a decided advantage in favour of the latter : —

1st, because they admit more readily of the diminution of h by
placing the centre of gravity lower.

2nd, on account of their greater mass.

3rd, on account of the greater distance between their centre of
gravity and the axles of the fore and hind wheels.

4th, on account of the increased value of k, the radius of gyration.

P. S. Since the above communication was read, an equally dis-
tressing event has taken place on the Great Western Railway, which
affords a striking illustration of the importance of these principles.
The persons who travelled upon or next to the luggage trucks were
the unfortunate victims on this occasion, and the fatal consequences ap-
pear to have arisen from the rolling of the trucks one over another, on
the train being unexpectedly stopped by a fall of earth lying upon
the railway. Had the buffers been placed at the proper horizontal
level, this rolling motion could not have taken place, and the loss of
human life might have been prevented.

J. POWER.

Trinity Hall,
April 12, 1842.

Vol. VII. Part III. M m

XVIII. Discussion of the Question: — Are Cause and Effect successive
or simultaneous? By the Rev. W. Whewell, B.D., Master of
Trinity College, and Professor of Moral Philosophy,

[Read March 14, 1842.]

I have at various times laid before this Society dissertations on the
metaphysical grounds and elements of our knowledge, and especially on the
foundations of the science of mechanics. As these speculations have not
failed to excite some attention, both here and elsewhere, I am tempted
to bring forward in the same manner some additional disquisitions of
the same kind. Indeed, the immediate occasion of the present memoir
is of itself an evidence that such subjects are not supposed to be without
their interest for the general reader; for I am led to the views and reason-
ings which I am now about to lay before the Society, by some remarks
in one of our most popular Reviews, {The Quarterly Review, Article on
the History and Philosophy of the Inductive Sciences. June 1841.) A
writer of singular acuteness and comprehensiveness of view has there made
remarks upon the doctrines which I had delivered in the " Philosophy of
the Inductive Sciences," which remarks appear to me in the highest degree
instructive and philosophical. I am not, however, going here to discuss
fully the doctrines contained in this critique. With respect to its general
tendency, I will only observe, that the author does not accept, in the
form in which I had given it, the account of the origin and ground of
necessary and universal truths. I had stated that our knowledge is de-
rived from Sensations and Ideas ; and that Ideas, which are the conditions of
perception, such as space, time, likeness, cause, make universal and necessary
knowledge possible ; whereas, if knowledge were derived from Sensation
alone, it could not have those characters. I have moreover enumerated a

M M 2

320 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

long series of Fundamental Ideas as the bases of a corresponding series of
sciences, of which sciences I have shown also, by an historical survey, that
they claim to possess universal truths, and have their claims allowed. I
have gone further: for I have stated the Axioms which flow from these
Fundamental Ideas, and which are the logical grounds of necessity and uni-
versality in the truths of each science, when the science is presented in the
form of a demonstrated system. The Reviewer does not assent to this
doctrine, nor to the argument by which it is supported ; namely, that Ex-
perience cannot lead to universal truths, except by means of a universal Idea
supplied by the mind, and infused into the particular facts which observa-
tion ministers. He considers that the existence of universal truths in our
knowledge may be explained otherwise. He holds that it is a sufficient
account of the matter to say that we pass from special experience to
universal truth in virtue of "the inductive propensity — the irresistible
impulse of the mind to generalize ad infinitum? I shall not here dwell
upon very strong reasons which may be assigned, as I conceive, for not
accepting this as a full and satisfactory explanation of the difficulty.
Instead of doing so, I shall here content myself with remarking, that
even if we adopt the Reviewer's expressions, we must still contend
that there are different forms of the impulse of the mind to generalize,
corresponding to each of the Fundamental Ideas of our system. These
Fundamental Ideas, if they be nothing else, must at least be accepted
as a classification of the modes of action of the Inductive Propensity, — >
as so many different paths and tendencies of the Generalizing Impulse :
and the Axioms which I have stated as the express results of the Fun-
damental Ideas, and as the steps by which those Ideas make universal truths
possible, are still no less worthy of notice, if they are stated as the results
of our Generalizing Impulse; and as the steps by which that Impulse,
in its many various forms, makes universal truths possible. The Gene-
ralizing Impulse in that operation by which it leads us to the Axioms
of geometry, and to those of mechanics, takes very different courses ; and
these courses may well deserve to be separately studied. And perhaps,
even if we accept this view of the philosophy of our knowledge, no
simpler or clearer way can be found of describing and distinguishing
these fundamentally different operations of the Inductive Propensity,

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 321

than by saying, that in the one case it proceeds according to the Idea of
Space, in another according to the Idea of Mechanical Cause; and the
like phraseology may be employed for all the other cases.

This then being understood, my present object is to consider some
very remarkable, and, as appears to me, novel views of the Idea of
Cause which the Reviewer propounds. And these may be best brought
under our discussion by considering them as an attempt to solve the
question, Whether, according to our fundamental apprehensions of the
relation of Cause and Effect, effect follows cause in the order of time,
or is simultaneous with it.

At first sight, this question may seem to be completely decided by
our fundamental convictions respecting cause and effect, and by the axioms
which have been propounded by almost all writers, and have obtained
universal currency among reasoners on this subject. That the cause must
precede the effect, — that the effect must follow the cause, — are, it might
seem, self-evident truths, assumed and assented to by all persons in all reason-
ings in which those notions occur. Such a doctrine is commonly asserted in
general terms, and seems to be verified in all the applications of the idea
of cause. A heavy body produces motion by its weight; the motion
produced is subsequent in time to the pressure which the weight exerts.
In a machine, bodies push or strike each other, and so produce a series
of motions ; each motion, in this case, is the result of the motions and
configurations which have preceded it. The whole series of such motions
employs time; and this time is filled up and measured by the series
of causes and effects, the effects being, in their turn, causes of other
effects. This is the common mode of apprehending the universal course
of events, in which the chain of causation, and the progress of time, are
contemplated as each the necessary condition and accompaniment of
the other.

But this, the Critic remarks, is not true in direct causation. " If the
antecedence and consequence in question be understood as the interpo-
sition of an interval of time, however small, between the action of the
cause and the production of the effect, we regard it as inadmissible.
In the production of motion by force, for instance, though the effect be

322 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

cumulative with continued exertion of the cause, yet each elementary
or individual action is, to our apprehension, instanter accompanied with
its corresponding increment of momentum in the body moved. In all
dynamical reasonings no one has ever thought of interposing an instant
of time between the action and its resulting momentum ; nor does it
appear necessary." This is so evident, that it appears strange it should
have the air of novelty ; yet, so far as I am aware, the matter has never
before been put in the same point of view. But this being the case,
the question occurs, how it is that time seems to be employed in the
progress from cause to effect? How is it that the opinion of the effect
being subsequent to the cause has generally obtained? And to this the
Critic's answer is obvious : — it is so in cases of indirect or of cumulative
effect. If a ball A strikes another, B, and puts it in motion, and B strikes
C, and puts it in motion, A's impact may be considered as the cause,
though not the direct cause, of C's motion. Now time, namely the time of
l?'s motion after it is struck by A, and before it strikes C, intervenes
between A's impact and the beginning of Cs motion : that is, between
the cause and its effect. In this sense, the effect is subsequent to the
cause. Again, if a body be put in motion by a series of impulses acting
at finite intervals of time, all in the same direction, the motion at the
end of all these intervals is the effect of all the impulses, and exists
after they have all acted. It is the accumulated effect, and subsequent
to each separate action of the cause. But in this case, each impulse
produces its effect instantaneously, and the time is employed, not in the
transition from any cause to its effect, but in the intervals between the
action of the several causes, during which intervals the body goes on with
the velocity already communicated to it. In each impulse, force produces
motion : and the motion goes on till a new change takes place, by the
same kind of action. The force may be said, in the language employed
by the Critic, to be transformed into momentum ; and in the successive
impulses, successive portions of force are thus transformed ; while in the
intervening intervals, the force thus transformed into momentum is
carried by the body from one place to another, where a new change
awaits it. " The cause is absorbed and transformed into effect, and therein
treasured up," Hence, as the Writer says, " The time lost in cases of

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 323

indirect physical causation is that consumed in the movements which take
place among the parts of the mechanism set in action, by which the
active forces so transformed into mechanism are transported over intervals
of space to new points of action, the motion of matter in such cases
being regarded as a mere carrier of force": — and when force is directly
counteracted by force, their mutual destruction must be conceived, as
the Reviewer says, to be instantaneous. We can therefore hardly
resist his conclusion, that men have been misled in assuming sequence as
a feature in the relation of cause and effect ; and we may readily assent
to his suggestion, that sequence, when observed, is to be held as a sure
indication of indirect action, accompanied with a movement of parts.

But yet if we turn for a moment to other kinds of causation, we
seem to be compelled at every step to recognize the truth of the
usual maxim upon this subject, that effects are subsequent to causes.
Is not poison, taken at a certain moment, the cause of disorder and
death which follow at a subsequent period? Is not a man's early pru-
dence often the cause of his prosperity in later life, and his folly, though
for a moment it may produce gratification, finally the cause of his
ruin? And even in the case of mechanism, if, in a clock which goes
rightly, we alter the length of the pendulum, is not this alteration the
cause of an alteration which afterwards takes place in the rate of the
clock's going? Are not all these, and innumerable other cases, instances
in which the usual notion of the effect following the cause is verified?
and are they not irreconcileable with the new doctrine of cause and effect
being simultaneous?

In order to disentangle this apparent confusion, let us first consider
the case last mentioned, of a clock, in which some alteration is made which
affects the i*ate of going.

So long as the parts of the clock remain unaltered, its rate will remain
unaltered ; and any part which is considered as capable of alteration, may
be considered as, if we please, the cause of the unaltered rate, by being
itself unaltered. But we do not usually introduce the positive idea of
cause, to correspond with this negation of change. If we speak of the
rate as unaltered, we may also say that it is so because there is no cause

324 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

of alteration. The steady rate is the indication of the absence of any cause
of alteration ; and the rate of going measures the progress of time, in a
state of things in which causes of change are thus excluded. If an altera-
tion takes place in any part of the clock, once for all, the rate is altered ;
but the new rate is steady as the old rate was, and, like it, measures the
uniform progress of time. But the difference between the new rate
and the old is occasioned by the difference of the parts of the clock ; and
the new rate may very properly be said to be caused by the change of
the parts, and to be subsequent to it : for it does prevail after the change,
and does not prevail before.

But how is this view to be reconciled with the one just quoted from
the Reviewer, and, as it appeared, satisfactorily proved by him ; accord-
ing to which all mechanical effects are simultaneous with their causes,
and not subsequent to them? We have here the two views in close
contact, and in seeming opposition.

In the going of a clock, the parts are in motion ; and these motions
are determined by forces arising from the form and connexion of the
parts of the mechanism. Each of the forces thus exerted at any instant
produces its effect at the same instant ; and thus, so far as the term cause
refers to such instantaneous forces, the cause and the effect are simul-
taneous. But if such instantaneous forces act at successive intervals of
time, the motion during each interval is unaltered, and by its uniform
progress measures the progress of time. And thus the motion of the
machine consists of a series of intervals, during each of which the motion
is uniform, and measures the time; separated from each other by a series
of changes, at each of which the change measures the instantaneous
force, and is simultaneous with it. And if, in this case, we suppose, at any
point of time, the instantaneous forces to cease, the succession of them
being terminated, from that point of time the motion would be uniform.
And since the rate of the motion in each interval of time is determined
by the instantaneous force which last acted and by the preceding motion,
the rate of the motion in each interval of time is determined by all the
preceding instantaneous forces. Hence, when the series of instantaneous
forces stops, the rate at which the motion goes on permanently, from that

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 325

point of time, is determined by the antecedent series of such forces, which
series may be considered as an aggregate cause ; and hence it appears, that
the permanent effect is determined by the aggregate cause ; and in this
sense the effect is subsequent to the cause.

Thus we obtain, in this case, a solution of the difficulty which is placed
before us. The instantaneous effect or change is simultaneous with the
instantaneous force or cause by which it is produced. But if we con-
sider a series of such instantaneous forces as a single aggregate cause, and
the final condition as a permanent effect of this cause, the effect is sub-
sequent to the cause. In this case, the cause is immediately succeeded
by the effect. The cause acts in time: the effect goes on in time. The
times occupied by the cause and by the effect succeed each other,
the one ending at the point of time at which the other begins. But the
time which the cause occupies is really composed of a series of instants
of uniform motion interposed between instantaneous forces ; and during
the time that this series of causes is going on, to make up the aggregate
cause, a series of effects is going on to make up the final effect. There
is a progressive cause and a progressive effect which go on together,
and occupy the same finite time ; and this simultaneous progression
is composed of all the simultaneous instantaneous steps of cause and
effect. The aggregate cause is the sum of the progression of causes;
the final effect is the last term of the progression of effects. At each
step, as the Reviewer says, cause is transformed into effect ; and it is
treasured up in the results during the intermediate intervals ; and the time
occupied is not the time which intervenes between cause and effect at
each step, but the time which intervenes between these transformations.

I have supposed forces to act at distinct instants, and to cease to act
in the intervals between ; and then, the aggregate of such intervals to make
up a finite time, during which an aggregate force acts. But if the action
of the force be rigorously continuous, it will easily be seen that all the
consequences as to cause and effect will be the same; the discontinuous action
being merely the usual artifice by which, in mathematical reasonings, we
obtain results respecting continuous changes. It will still be true, that
the uniform motion which takes place after a continuous force has acted, is
the effect subsequent to the cause; while the change which takes place
Vol. VII. Part III. N n

326 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

at any instant by the action of the force, is the instantaneous effect simul-
taneous with the cause.

It may be objected, that this solution does not appear immediately
to apply : for the motion of a clock is not uniform during any portion
of the time. The parts move by intervals of varied motion and of rest ;
or by oscillations backwards and forwards; and the succession of forces
which acts during any oscillation, or any cycle of motion, is repeated
during the succeeding oscillation or cycle, and so on indefinitely ; and if
an alteration be made in the parts, it is not a change once for all, but
recurs in its operation in every cycle of the motion.

But it will be found that this circumstance does not prevent the same
explanation from being still applicable with a slight modification. In-
stead of uniform motion in the intervals of causation, we shall have to
speak of steady going: and instead of considering all the forces which
affect the motion as causes of change of uniform motion, we shall have
to speak of changes in the parts of the mechanism as causes of change
of rate of going. With this modification, it will still be true, that
any instantaneous cause produces its instantaneous effect simultane-
ously, while the permanent effect is subsequent to the change which
is its cause. The steady going of the clock is assumed as a normal
condition, in which it measures the progress of time ; and in this assump-
tion, the notion of cause and effect is not brought into view. But a
steady rate thus denoting the mean passage of time, a change in the
rate indicates a cause of change. The change of rate, as an instantaneous
transition from one rate to another, is simultaneous with the change
in the parts. But then the changed rate as a continued condition in
which, no new change supervening, the rate again measures the progress
of time, is subsequent to the change of parts, for it begins when that ends,
and continues when the progress of that has ceased.

If, however, this be a satisfactory solution of the difficulty in the case
of mechanism, how shall we apply the same views to the other cases ?
Growth, the effect of food, is subsequent to the act of taking food ;
disorder, the effect of poison, is subsequent to the introduction of poison
into the system. Can we say that the animal would continue unchanged

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 327

if it were not to take food ; and that food is the cause of a change,
namely, of growth ? This is manifestly false ; for if the animal were
not to take food, it would soon perish. But the analogy of the former
case, of the clock, will enable us to avoid this perplexity. As we assumed
a steady rate of going in the clock to be the measure of time when
we considered the effect of mechanism, so we assume a steady rate of
action in the animal functions to be the measure of the progress of
time when we consider the causes which act upon the development
and health of animals. Digestion, and of course nutrition, are a part
of this normal condition ; they are involved in the steady going of the
animal mechanism, and we must suppose these functions to go regularly
on, in order that the animal may preserve its character of animal. Food
and digestion may be considered as causes of the continued existence of
the animal, in the same way in which the form of the parts of a clock
is the cause of the steady going of a clock. And when we come to
consider causes of change, this kind of causation, which produces a normal
condition of things, merely measuring the flow of time, is left out of our
account. We can conceive an uniform condition of animal existence, the
animal neither growing nor wasting. This being taken as the normal condi-
tion, any deviation from this condition indicates a cause, and is taken as the
evidence and measure of the cause of change. And thus, in a growing animal,
the food partly keeps the animal in continued animal existence, and partly,
and in addition to this, causes its growth. Food, in the former view, is
always circulating in the system, and is supposed to be uniformly adminis-
tered ; the cycles of nutrition being merged in the notion of uniform
existence, as the oscillations of the pendulum in a clock are merged in
the notion of uniform going ; and the elementary steps of nutrition which
are, in this view, supposed to take place at each instant, produce their in-
stantaneous effect, for they are requisite in the cycle of animal processes
which goes on from instant to instant. But on the other hand, in con-
sidering growth, we compare the state of an animal with a preceding
state, and consider the nutriment taken in the intervening time as the
cause of the change : hence this nutriment, as an aggregate, is considered
as the cause of growth of the animal; and in this view the effect is
subsequent to the cause. But yet here, as in the case of mechanism,

N N2

328 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

the progressive effect is simultaneous, step by step, with the progressive
eause. There is a series of operations ; as for instance, intussusception,
digestion, assimilation, growth : each of these is a progressive operation ;
and in the progress of each operation, the steps of the effect and the
instantaneous forces are simultaneous. But the end of one operation
is the beginning of the next, or at least in part, and hence we have
time occupied by the succession. The end of intussusception is the be-
ginning of digestion, the end of digestion the beginning of assimilation,
and so on. These aggregate effects succeed each other; and hence
growth is subsequent to the taking of food ; though each instantaneous
force of animal life, no less than of mechanism, produces an effect
simultaneous with its action. Each of these separate operations is an
aggregate operation, and occupies time; and each aggregate effect is a
condition of the action of the cause in the next operation.

Again ; if an animal in a permanent condition, neither waxing nor
wasting, may be taken as the normal state in which the functions of life
measure time, in order that we may consider growth as an effect, to be
referred to food as cause; we may, for other purposes, consider, as the
normal condition, an animal waxing and then wasting, according to the
usual law of animal life : and we must take this, the healthy progress of
an animal, as our normal condition, if we have to consider causes which
produce disease. If we have to refer the morbid condition of an animal
to the influence of poison, for example, we must consider how far the
condition deviates from what it would have been if the poison had not
been taken into the frame. The usual progress of the animal func-
tions including its growth, is the measure of time; the deviation from
this usual progress is the indication of cause; and the effect of the poi-
son is subsequent to the cause, because the poison acts through the
cycle of the animal functions just mentioned, which occupies time ; and
because the taking the poison into the system, not any subsequent
action of the animal forces in the system, is considered as the event
which we must contemplate as a cause. To resume the analogy of the
clock : the rate of the clock is altered by altering the parts ; but this
alteration itself may occupy time ; as if we alter the rate of a clock
by applying a drop of acid, which gradually eats off a part of the

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 329

pendulum, the corrosion, as an aggregate effect, occupies time; and
the rates before and after the change are separated by this time. But
the application of the drop is the cause; and thus, in this case the final
effect is subsequent to the cause, though here, as in the case of me-
chanism, the instantaneous forces always produce a simultaneous effect.

Thus we have in every case a uniform state, or a state which is
considered as \xniform, or at least normal; and which is taken as the
indication and measure of time; and we have also change, which is con-
templated as a deviation from uniformity, and is taken as the indication
and measure of cause. The uniform state may be one which never
exists, being purely imaginary ; as the case in which no forces act ; and
the case in which animal functions go on permanently, the animal neither
growing nor wasting. The normal state may also be a state in which
change is constantly taking place, as, in fact, even a state of motion is
a state of change; such states also are, in a further sense, that of a
clock going by starts, and that of an animal constantly growing : in
these cases the changes are all merged in a wider view of uni-
formity, so that these are taken as the normal states. And in all these
cases, successive changes which take place are separated by intervals of
time, measured by the normal progress ; and each change is produced by
some simultaneous instantaneous cause. But taking the cause in a larger
sense, we group these instantaneous causes, and perhaps omit in our
contemplation some of the intervening intervals ; and thus assign the
cause to a preceding, and the effect to a succeeding time.

I may observe further, as a corollary from what has been said, that
the measure of time is different, when we consider different kinds of
causation ; and in each case, is homogeneous with the changes which
causation effects. In the consideration of mechanical causes, we measure
time by mechanical changes; — by uniform motion, or uniform succession
of cycles of motion ; by the rotation of a wheel, or the oscillation of
a pendulum. But if we have to consider physiological changes, the
progress of time is physiologically measured; — by the normal progress
of vital operations ; by the circulation, digestion or developement of the
organized body; by the pulse, or by the growth. These different measures
of time give to time, so far as it is exhibited by facts and events,

350 PROFESSOR WHEWELL, ON CAUSE AND EFFECT.

a different character in the different cases. Phenomenal time has a dif-
ferent nature and essence according to the kind of the changes which
we consider, and which gives us our sole phenomenal indication of
cause.

I fear that I am traveling into matters too abstruse and metaphysical
for the occasion : but before I conclude, I will present one other aspect
of the subject.

In stating the difficulty, I referred to cases of moral as well as physical
causation ; as when prudence produces prosperity, or when folly produces
ruin. It may be asked, whether we are here to apply the same ex-
planation ; — whether we are to assume a normal condition of human
existence, in which neither prudence nor folly are displayed, neither pros-
perity nor adversity produced ; — whether we are to conceive the progress
of such a state to measure the progress of time, and deviations from it
to denote causes of the kind mentioned. It may be asked further, whether,
if we do make this supposition, we can resolve the influence of such causes
as prudence or imprudence into instantaneous acts, which produce their
effects immediately : and which occupy time only by being separated by
intervals of the inactive normal moral condition. To this I must here
reply, that the discussion of such questions would carry me too far, and
would involve speculations not included within the acknowledged domain
of this Society, from which I therefore abstain. But I may say, before
quitting the subject, that I do not think the suppositions above suggested
are untenable ; and that in order to include moral causation under the maxims
of causation in general, we must necessarily make some such hypothesis.
The peculiarity of that kind of causation which the will and the character
exert, and which is exerted upon the will and the character, would make
this case far more complex and difficult than those already considered ; but,
at the same time, would offer us the means of explaining what may seem
harsh, in the above analogy. For instance, we should have to assume
such a maxim as this : that in moral causation, time is not to be measured
by the flow of mechanical or physiological events ; — not by the clock, or
by the pulse. Moral causation has its own clock, its own pulse, in the
progress of man's moral being ; and by this measure of time is the relation
of moral cause and effect to be defined.

PROFESSOR WHEWELL, ON CAUSE AND EFFECT. 331

That in estimating moral causation, the progress of time is necessarily
estimated by moral changes, and not by machinery, — by the progress of
events, and not by the going of the clock, — is a truth familiar as a practical
maxim to all who give their thoughts to dramatic or narrative fictions.
Who feels any thing incongruous or extravagantly hurried in the progress of
events in that great exhibition of moral causation, the tragedy of Othello?
If we were asked what time those vast and terrible and complex changes
of the being and feelings of the characters occupy, we should say, that,
measured on its own scale, the event is of great extent ; — that the trans-
action is of considerable magnitude in all ways. But if, with previous
critics, we look into the progress of time by the day and the hour —
what is the measure of this history? Forty-eight hours.

But I am going beyond the boundaries of the speculations which we
usually follow in this room, and will conclude.

W. WHEWELL.

XIX. On the Motion of a small Sphere acted upon by the Vibrations of
an Elastic Medium. By the Rev. James Challis, M.A., Plumian
Professor of Astronomy in the University of Cambridge.

[Read April 26, 1841.]

It is proposed in this Essay to give a mathematical investigation re-
specting the motion of a small solid sphere submitted to the dynamical
action of the vibrations of a medium so constituted that the pressure {p)
and density (p) are related to each other by the equation p = ofp, a 2 being
a certain constant.

1. For this purpose it will be convenient to obtain, first, the equations
which apply to the motion of such a medium when directed to or from a
centre, whether the centre be moving or stationary.

Conceive P to be a fixed point in space at which the motion of the fluid
is directed to or from a moving centre C. Describe about C as a centre a
spherical surface always passing through the point P, and concentric with
this another passing through P', a point in CP produced. Let, at a given
time t, CP = r, and CP' = r, or r + $r, $r being supposed very small.
Conceive now a conical surface, with an indefinitely small vertical angle, to
have its vertex at C, and its axis coinciding with CPP, and let it always
include the same portion (*»*) of the interior spherical surface. Then if
a = the velocity of the centre C resolved in the direction of r, the radius
CP at the time t + t, (t being very small) will become r ± ax, and CP'
will become r + Ir ± ar, the interval $r being supposed not to vary with
the time. Hence the portion of the exterior surface included by the

Vol. VII. Part III. Oo

334 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

conical surface at the time t + r is m z , ( =—I ) or /» 2 . ( 1 + —

V r ± nx J V r ± ar

a Jm

and this, neglecting terms of the order Sr x a-r, is equal to — - .

Again, let v and p be the velocity and density of the fluid which passed
the area m 2 at the time /, and v /t p , the values of the same quantities at any
time t + t. Now the quantity of fluid which in the small time dt passes m %
18 equal to jm i p l v i dT, the integral being taken from t = to t = St. And

because

d.pv ,

pv t = pv + ~~jr- ~r very nearly,

this integral is equal to

, * , d.pv ^ty

nfpvSt + m- . — ~ . ±~ + &c.

Also if v], p[ be the velocity and density of the fluid which is passing the

area — — of the exterior surface at the time t + t, the quantity of fluid
r

— j— vlp'dr, taken from r = to r = St.

And, because v] and p,' are what v and p become by very small changes of
time and place,

, , d.vp d.vp*

Hence,

/•jbV" , , 7 m 2 / 4 r . d.vp d.vp, .'

r 4 l r dt 2 dr

Consequently, supposing the velocity positive when directed from the
centre, the increment of matter in the space between the two areas in the
time St, is ultimately,

~ ^r^ v P + C -dT lr) ~ "* v p}* e '>

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 385
or, putting v'p for vp + —7— Sr,

dr

m*St.

1

r~v p — rvp

Now if any point be selected between P and P, the radius to which at
the time t is r t , by what has been already shewn, the transverse section of
the cone through this point at the time t + St is with sufficient approxima-

»«V, 2

tion — -'- , and is therefore independent of St. Hence at any instant
during the interval It the content of the conical frustum is • f—jr dr,

'ill*

{from r t = r to r t = r'\, or — (r' 3 - r»). The increment of density (Sp)
in that space in the time St is consequently,

m 2 St(r'°p'v - r'pv) 3r

s

r~ m? (r' 3 — r 3 ) '

Hence, fe + ? (^ ~f> ) - „
d£ r 3 — r 3

and passing from differences to differentials,

&-33£*«~r f <"■

It is plain that since i 3 has been assumed to be a fixed point of
space, the differential coefficients here are partial. The above equation,
with

P = <*P (2),

and that derived from D'Alembert's Principle, viz.

£♦<&-■ »

are the three eqviations which determine the circumstances of the motion.
As the velocity (a) of the centre C in no way enters into them, we
may conclude that the same equations apply to motion tending to or from
a moving centre as to motion tending to or from a fixed centre.

002

336 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

2. From the equations (1), (2), (3), others more immediately appli-
cable to the question proposed to be discussed will now be deduced.

The equation (1) is equivalent to
dp dv vdp 2v

W;

pdt dr pdr r

and by substituting for p in (3) from (2) there results,

a' dp dv vdv
Jt

;d7 + ( rt + -dT = ° < 5) -

If now we assume <p' to be such a function of r and t that the
partial differential coefficient -T~ is equal to v, and substitute this ex-
pression for v in (5), the equation is integrable with respect to r. The
result is,

* Nap. log. P + *£ + ^ =f(t).

To get rid of the arbitrary function of the time, suppose

$'=<!> + ff(t)dt.

Then ^ = ^4-/(0, and^' = ^t Hence
dt at ° v ' dr dr

" n "p- •* * + ft + ^ - ° < 6 >

Obtaining from this equation — j and — p- , substituting their values in

(4), and putting ~ for v, the result will be,

d*<p l d<p \ 1 d'(p 2 d<p d 2 <j> 2^ = ,

dr 2 \ (fdr*) a* dt 1 a? dr drdt r dr

3. Before making use of this equation it will be necessary to
consider the comparative values of its terms under the circumstances
in which we propose to apply it. The circumstances are, that v is
very small compared to a, and r always exceedingly small compared
to the breadths of the waves whose dynamical action is to be inves-
tigated.

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 337

First, it is plain that the terms having or in their denominators
will be small compared to the others. Neglecting those terms, or,
which is the same thing, considering a infinite, we have the case of
an incompressible fluid, and the equation applicable to it is,

d'<p d(p d*- r( P_ n

dr' ! ' ' rdr dr*

The integral of this equation is,

Hence d -± -*& t and % = -££ + F>(t).
dr r* dt r v J

The known equation which gives the pressure (p) of an incom-
pressible fluid is

d(b v* n

Hence by substitution,

•*& - 4 - »»

As this equation contains two arbitrary functions, two conditions of
the motion may be arbitrarily assumed. Let us assume for one con-
dition, that the excess of the pressure (jo) above the pressure n, which
would exist in the undisturbed state of the fluid, is solely owing to
a velocity arbitrarily impressed in the direction of r. Then v and A f{t)
being supposed to vanish when p = IT, we must have,

p — n = - — — — .

As a second condition, let us suppose that the velocity is impressed
at a given distance (r), and is given at any time t by the expression
m sin bt. Hence f(t) = mr> sin bt, and f(t) = bmr- cos bt. Consequently
by substituting,

p - n = mbr cos bt — — sin* bt,

:J38 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

an exact equation, which gives the pressure at the distance r at any
time. The two terms will be of the same order if m be not very
small compared to 2br.

Next, let us try the effect of retaining one of the omitted terms
of the equation (7) and neglecting the others. Retaining, first, the

term -£ x „ , „ , and putting v for -^ , we shall have,
dr" adr 2 l ° dr

dv t v~\ 2v _

dr \ a 2 ) r

dv vdv 2

or — —j- + - = 0.

vdr a'dr r

Integrating with respect to r,

v*
Nap. log. vr ! - -3 = Nap. log./(0 ;

.-. e*» =f(t) A = /'(/) (1 + £ + &c.)

Hence neglecting terms removed in order by two degrees from those

fit)
retained, v = ^—^ • This is the same result as in the case of the in-
r

compressible fluid, and by reasoning in the same manner as for that

case, it will be found from equation (6) that

o«Nap. log. P =-^-£- **(/).

If p = 1 + o-, and we assume that the value of a depends only on
a disturbance in the direction of r, it will follow that F' (t) = 0, v and
,f'(t) being supposed to vanish when p — 1.

Hence P = e* r ™ = 1 +«-^ ; , nearly,

J 2 fit) V* ,

and (i"cr =' / —^-L nearly.

r 2 J

It appears, therefore, by the foregoing reasoning, that whenever

— J, is of the same order as t> 2 , the first term introduced into the
r

expression for the pressure by the term of equation (7) which has now
been considered, is of the order of v\

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 339

Again, let us retain the term —■-?-■ , ,.. , rejecting the others. Then

d*<j> _ 2 dj> d 2 <f> 2dcj>

dr 11 a 2 ' dr ' drdt rdr ~ '

dv 2 d 2 (p 2
tar a 2 ' drdt r

Hence, integrating with respect to r,

2 d<t>
Nap. log. vr 2 - - 2 . -^ = Nap. log./(0;

.-. vr 2 = f(t) .&% =f(t) • (l + |. -£) , nearly.
Under the same conditions as in the last case, the first approximations

to the value of v and -3? are ^-^ and - — — . Substituting- this latter

at r* r &

quantity in the above equation,

dr r* «y '

and integrating with respect to r, without adding an arbitrary function
of the time,

d,= /(*> . /CO/'**) .

v r d'r* '

.-. ** = _•££> . l /'(0}'+/(0/'(0
flfa r aV-

To take a particular instance, let the velocity impressed at the time t
at the distance r from the centre, and in the direction of this radius,

be m sin — - — , a being supposed very large compared to m, and X very

large compared to r. Then, for first approximations,

, Vj ,. , . 2Trat „,.. 2-namr 2 9,-n-at , „,,.. 4nrWmr- . 2ttuI
f (t) = m r 2 sin -y- ,f(t) = — — - cos — , and/ (0 - sin -^—

These values will enable us to estimate the order of the second term of
the above expression for -j- . By substitution they give for this term,

340 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

4Tr-'/«V iirat , . , ... j , , • e ., , „ m~- r*

- „ — cos , which, with regard to «- is of the order of — - x — ,

XX a X

that is, of the fourth order; whilst the first term in the expression for

-r- is with regard to a of the order of — x — , that is, of the second
dt fo a X

order. Hence we may conclude as before that the first term introduced

into the expression for the pressure by the term of equation (7) just

considered, is of the fourth order.

4. Lastly, let us retain the term ; . -, ™ of equation (7), and

reject the other small terms. We shall then have,

#£ 1 cP$ 2 d±_ Q . Qr d\r<f> d\r<p

The known integral of this equation is

r<p =f(r - at) + F(r + at).

The second arbitrary function applies to a disturbance which causes
propagation towards the centre; and as such a motion is excluded by
the nature of the question to the solution of which the present reasoning
is directed, I shall suppose this function to vanish. Then,

f(r - at)
<*> = - r •

d< t> - „ f'( r - at )
dt r

d£ _ fir -at) f(r - at)
dr ' r r 2

As an application of this solution, let us suppose the velocity impressed
at any time t in the direction of r, and at the distance r from the centre,
to be m(p(t). Then, putting for shortness' sake u for f{r — at), we
shall have

1 du u
r ar dt r

du a .,-.■«

or —. — h - . u 4- marcb (t) = 0,
at r

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 341

an equation in which u and / are the only variables, and which serves
to determine the value of the function f{r — at) from the given value
of <j>(t). This equation gives by integration,

_at _at at

u = Ce r — mare r fe r <p(t)dt,

du Ca — — — — —

-r. = e r - mare r fe r <p'(t)dt.

Now -j- = - a f'(r - at) = r-£. Therefore

deb Ca -% -^ r- ,

dt = ~~r^ e "- mae r fe r <p\t)dt (8),

As an example of the application of this formula, let us suppose
as before that <p{t) = sin ^_* . Then 0'(/) = *^cos— '. Also

e cos — - — dt =

T 2tt«/ _, \r XX X

7 (a 2irat Zira . 2nat\

eM-.cos — — — — sin

r* + \*

at

\e r . (2-n-at

.sin i

2ira

(%-n-at \
a COS (_ - «) ,

by substituting tan a for . Hence

rf0 Ca -g . /27ra#

= «■ e — 7»a sin i

a*rf

r

.2

a COS ( — a 1 .

and consequently by equation (6),
«<Nap. log.p = Ca e -1 + Hasina cos (!=2f - a) - |%in*^ (9).

It will be seen by the above result that the term of equation (7),
retained in this instance, introduces into the expression for p a quantity

of the order of — x sin 2 a, or of - x r?, and therefore of the third order,
fit o, \

Hence that term is more considerable than the other small terms of

equation (7), and we may be confident that by retaining it and rejecting

Vol. VII. Part III. P P

342 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

the others, our approximation will be exact to at least terms of the
second order.

To determine the arbitrary constant in the equation above, let us
suppose that when t = 0, p = 1. Hence

Ca

—t- + ma sin a cos a = ;
r

j 2 xt i f ~T . (2-n-at \) m* . n %wat
and a Nap. log. p = ma sin a I — cos o e r + cos I a > — — sin- .

It is worthy of remark, that if we confine ourselves to quantities of
the second order, the above result coincides with that obtained in Art. 3,
for an incompressible fluid. For to that degree of approximation

2ttT

sin a = — — ; and if p « I + <r, a: Nap. log. p = a* a. Also the term in-

at

volving e~ T will disappear after a very short time on account of the

great magnitude of -. Hence

„ ■ 2ira Qirat m? . .Qicat

are = m . . r cos — sin* — - — ,

X X 2 X

which, by putting b for , evidently coincides with the result ob-

X

tained in Art. 3.

5. Prior to the consideration of the dynamical action of the fluid in
vibration on a small sphere, it will be convenient to determine the pressure
on the surface of a small sphere performing small rectilinear vibrations in
the fluid at rest.

The sphere is supposed to be perfectly smooth, and therefore incapable
of impressing motion on the fluid in directions perpendicular to the radii.
Hence the motion given to the fluid by the motion of the sphere is
directed to or from a moving centre. If V be the velocity of the sphere at
the time t, and 9 the angle which a radius to any point of the surface makes
with the straight line in which the centre is moving, VcosO is the velocity
impressed on the fluid at that point at the same time. Now as this normal
velocity varies at a given instant from one point to another of the surface,

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 343

it follows that there will also be variation of density. The effect of this
variation of density will be to cause the motion of each particle in contact
with the spherical surface to be curvilinear, and to be continually directed
to or from the varying positions of the centre*. Hence the equation (1)
obtained in Article (1) will be applicable to the case before us by merely
substituting F'cos 9 for v. The same substitution being made in equation
(3), the three equations (1), (2), (3) may be immediately made use of for
our present purpose.

Let V = mcp(t). Then (8) becomes for this case,

dd> Ca -Si -^1 r a J.

-j7 = z~ e T ~ am cos ® e ' l er <P'(t)dt,

and from equation (6),

f A at at at

a 2 Nap. log. P = — T e~ ' + am cos 9e'~ fe T <p'(t) dt - m 2 cos 2 6 ^j] 2 -

/' — il

e r (p'{t)dt = e r \// (t), and that p = 1, <p (t) = 0, and

xj/ (t) = k, when t — 0. Hence,

f

= — - + kam cos 6,

r l

J -2il <m 2

and « 2 Nap. log. P = am cos 9 {$ (t) - he r \-~z- cos 2 9 ^>(t)]\ (10.)

So

If we put 1 + er for p and neglect terms of the order of o- 2 , we shall
have a 2 Nap. log. p = a* a = the effective pressure on a unit of surface
of the sphere. The effective pressure on the whole sphere estimated in
the positive direction of the sphere's motion is — l^r* fa 2 a sin 6 cos 6d0,
taken from 9 = to 8 = tt. The negative sign is prefixed because it has
been already assumed that the velocity of the fluid is positive when it tends
from a centre, and as the central velocity in this instance is m cos 9<p(t),
9 must be measured from the point of the sphere which is foremost when
the motion is in the positive direction, so that the resultant of the pressure
on an annulus of breadth rd9 and radius r sin 9 is in the negative direc-
tion when cos 9 is positive.

* See the proof of this assertion in the * Note ' added to this paper.

PP2

344 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

Now since / sin 6 cos 2 dO = -, and I sin 9 cos* 0d9 = 0, the whole

resulting pressure on the sphere is

47r«mr 8 i. ... 7 -£l
3 VMO -** J-

And if 5 be the ratio of the density of the fluid to that of the sphere, the
accelerative force of the resistance of the fluid is

-Si

which on account of the very small factor e r will after a short interval
become,

.—■+«>•

6. Suppose, for example, the sphere to vibrate as a pendulum, and the
extent of the vibrations to be so small that the motion of the centre may
be considered rectilinear. Let / be the length of the pendulum, and x the
distance of the centre of the sphere at the time t from the position it would
have at rest. Then, taking the buoyancy of the fluid into account, we

have for the accelerative force of gravity — — - (1 - $)•, and consequently,

by the foregoing reasoning,

d 2 x gx . «, amS -ili r zl d*x .. ,1

df I ' r \J mdf* J

_ T r a 4d*x At Zfr d*x r % d'x\ . _ , ,

Now J e w dt = e {a-dr-a'-d?) yer y nearl y* Hence ' b ^ sub -

stituting,

at

d*x _gx 1 — $ rS d 3 x kam§

df ~ T'TTS + a{l+8)"3F + r(\+$) e ' { >'

Hence, for a first approximation, after a very small time,

d 2 x ffx /1 — S

d'x _ gx /l — 6\
df~~~T' [l+li

This equation not containing a is true of an incompressible fluid. It
is, in fact, when applied to this case, an exact equation, as appears

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 345

from the reasoning in the Cambridge Philosophical Transactions, Vol. V.

Part ii. p. 200. By equating the factor in brackets to 1— n<>, it will

2
be seen that »= - — r, which when S is very small is nearly equal to 2*.

1 + o

Before proceeding to a second approximation let us determine the
value of the constant h. This will be obtained by finding the value

f Cm*" "V tr il^ 7*

of -. , -■ y—j-z> when £ = 0. Now from (11), neglecting the term

it Tit itt it' fw it t

involving - as a factor, and differentia ting s

at

('

d s x _ g dx \ — l ka?m§
W ~~ T"dt *T+7 ~ ^(1 + ty

and from the same equation,

d 2 x _ r cPx gx 1 — 8 r e^r kamh - <£

df ad? T'l+S ~ a{l + 8)"dF + r(T+7j * ''

Hence, supposing x = h and j- =0 when t = 0, we readily obtain,

Whence,

kam gh 1 — 5 kam$ kaml

r = ~ T * 1+7 + r(l+5) 2 + RT+1)

kam _ gh n *s

d 3 x

Substituting now this value in the approximate expression for -r— ,

we get

d*x_ g dx l-S ghat °J-

d? -~l'dI'TT$ + ~iT- {1 - S)e '

* I have already obtained this result in the London and Edinburgh Philosophical Mag-
azine for September 1833 (p. 186), in the Cambridge Philosophical Transactions as above
cited, and more recently in the Philosophical Magazine for December 1840 (p. 46l). The
reasoning in the last of these solutions, not embracing those terms involving the square of
the velocity which may be of equal magnitude with terms retained, cannot be considered so
complete as that I have now given.

346 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE
and consequently by (11),

tff = ~ £ "l+a /a'(l + 3) 2 '</* T'm ,e '■

Putting for shortness' sake n 2 for *•- — j, we shall have

d*x n*r$ dx , „>. -i

-=- + ,. • -T- + w 2 « = - rc 2 AS<? r .

a£ 2 « (1 + a) a£

r*

By integrating this equation and neglecting terms involving — ,

which will be wholly insignificant, it will be found that

dx - " M '-

— - = — hue 2 "( 1+s >. sin nt,

at

by means of which equation the decrements of the successive arcs of
vibration may be calculated. It is remarkable that for an incompress-
ible fluid, for which a is infinitely great, there is no decrement of
the arcs excepting so far as it arises from friction and capillary at-
traction. The index of e in the equation above is too small to account
for the observed decrements in air, which must be mainly owing to
friction.

7. As another example, let the velocity = m(Z(l — e~ yt ) + m<p(t)
Then

e'^(t) = ffi Wye-*' + (j>'(t)} dt = -&- e^~^' + fe r '<p'{t) dt.

r

Hence f (/) = EZ + e r fe r <p' (t) dt,

a

r '

a

7

r

and the accelerative force of the resistance is

ami f /3 7 v . ~£ r a T^w*\j* i "
I s ' e~ yt + e r J e r (p (f) dt — ke

£-*

If 7 be an exceedingly large quantity, in which case the sphere's
velocity after a very short interval is m {/3 + <p(t)}, the above result
becomes for all values of t which are not exceedingly small,

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 347

e ' fe r <p'(t)dt,

and therefore the same as if the velocity were simply m<j>(t). Hence
a small sphere moving with a uniform velocity suffers no resistance;
and if its velocity be partly uniform and partly variable, the resistance
depends only on the variable part.

9. I come now to the consideration of the motion of a small sphere
supposing it acted upon by the pressure resulting from a series of
vibrations of the fluid, no other force acting. I suppose the vibrations
to be propagated with the uniform velocity a in the positive direction,
and the velocity of the vibrating fluid to be m<p(t) at the origin of
x -at any time t. At the same time t at any distance x from the

origin the velocity is <p [t ■) , being that which was at the origin

at the time it ) . Suppose the centre of the sphere to be at the

origin of x ' when t = 0, and to be at the distance x at the time t.
For the sake of simplicity I shall first assume the vibrations of the
fluid to be unaccompanied by change of density, which is a supppsable
case if we conceive all the parts of the fluid to move in the direction
of x at the same time with the same velocity. Now it is clear that
the action of the fluid on the sphere depends only on the difference
of their velocities. And the mathematical conditions of the question
will remain the same if we suppose the fluid to be at rest and the

sphere to have the velocity — \mfpit ) — -tt } • Calling this velocity

— mx(t), and f the resulting accelerative force of the sphere, we shall
have by what was proved in Art. 5,

- ami \ -2* r ii -al\

J= ~-\e T Je'^{t)dt-ke ').

Let us now suppose the fluid vibrations to be accompanied by change

of density. If pi be the density where the velocity is m<p It J , it

-<t>(t- x \ The
is well known that we have the exact relation pi = e a N «/'

348 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

velocity of the fluid being supposed to remain the same as before, the
effect of change of density will be taken into account by merely sub-
stituting p^ for 5 in the equation above. By this substitution let f
become f. Then

f = ^le^ t) + ^{e^ fe-^[t)dt-ke^}.

Again, the sphere will be acted upon by an additional accelerative force

arising from the circumstance that the density, and consequently the

pressure, varies from one point to another of its surface at a given

time, on account of the variation of density of the fluid in vibration

with the distance x from the origin at a given time. The pressure

at all points of a plane perpendicular to the direction of x will evidently

be the same. Hence, if c?f(x) represent the pressure at any distance x,

corresponding to the position of the centre of the sphere, it may readily

d f(x)
be shewn that the accelerative force in question is — a 2 3— H^-^, terms

involving r 2 being omitted. Now

f(x) = e a ' , and — ^M = : • « "' * <t> [t .

* ax a 2 \ a)

Hence, calling this force f", we have,

As the sphere is solicited by no other forces than those just considered,

d 2 x
f +/" = -Tj ; and consequently,

at

Now fe' x'tydt = «' '- { x '(t) - - X "(t)}> nearly:

U fJh

and e a = e" v a/ a< " = e ffl «-x I — 1 , nearly. Hence

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 349
Consequently, by integrating to the second approximation,
c-a.e a ** = aU" y "' + aSe a x O + ***** + c x(0-

a ™ x a

When #, £, and -7- each = 0,

and to the same degree of approximation k = - x'(0)- Hence c = a + 2a&.
Therefore, when t is not exceedingly small,

a + 2«S - ae-™ = aiG-* (# ' S + e- x, °) - — *'(*),
and expanding the exponentials to terms containing m 2 ,

dx dx*
dl ~ lad?

'"*{+{'- a) + XWI + **{+('- a) + X(0l)-— X(0-
Hence -7- = x <p (t ) , for a first approximation.

Therefore x «) - f (« - S) (l - jMj) - # (l - 2). £| .

to the same approximation.

^»'W-»(«-S-m-(»-a)->('-S)-T^'-»-

*- Bf'* --*♦(« -3*= [^F3lM 1+ S^)1]

mrZ 1 — J" , /. x\
And finally,

dl = TTy * V ~ a) ~ «(!+*)■ + V - a) + £(T73j ' * V ' a) '

I shall content myself in the present communication with having
obtained this equation, which, as far as I am aware, is the first instance
of a solution of a problem of this kind. On a future occasion I propose

Vol. VII. Past III. Qq

350 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

making some applications of it. I shall here only remark, in confirmation

of the result, that if U J =1, that is, if the fluid move with the

.- , .. dx 2mh m 2 ^ , ^, .

uniform velocity m, -j- — z — * H 7^ — -rt ; and supposing the density

of the sphere to be the same as that of the fluid in motion, and con-

-- dx

sequently 8 = e " , it will be found that —r- = m, neglecting m s , &c.

This manifestly should be the case. If

2tt

* [ a) = Sin ¥ * " )'

dx
the last term of the expression for «*- will be partly constant and partly

variable, and it is plain from equation (12) that the accelerative force

is the same as if the constant part did not exist. If <p' (t ] =0,

whenever <p (t J =0, the sphere and the fluid will be stationary at

the same instants.

J. CHALLIS.

Cambridge Observatory,
March 3, 1841.

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 351

ADDITIONAL NOTE.

The theoretical resistance to the motion of a ball-pendulum in the
air, obtained in the foregoing Essay, differs from that found by Poisson
in Vol. xi. of the Memoirs of the Paris Academy of Sciences and in the
Connaissance des Terns for 1834, and by M. Plana in a Memoir on the
Motion of a Pendulum in a Resisting Medium, published at Turin in 1835.
I propose therefore to add here as distinct a statement as possible of the
reason of this difference.

According to the solution of these two eminent mathematicians, the
motion of the fluid in contact with the oscillating sphere, is partly along
its surface and partly directed to or from the centre : on the contrary,
in the solution I have given, the motion at each instant is wholly di-
rected to or from the centre. The following reasoning appears to prove
the correctness of the latter view.

It is well known that the equation udx + vdy + wdz = 0, is the
differential equation of a surface which cuts at right angles the directions
of the motions of the particles through which it passes, if the left hand
side of the equation be integrable per se. And if it be integrable after
being multiplied by a factor N, the equation N(udx + vdy + wd%) = 0,
is equally the differential equation of such a surface, but more general
in its application. Let therefore

N (udx + vdy + wdz) — d. (p (x, y, z, t).

Then integrating, and supposing the arbitrary function of the time to
be included in cp, we shall have (x, y, z, t) — 0. The surfaces of which
this is the general equation, for the sake of shortness I shall call surfaces
of displacement. If the time t changes to t + dt, the co-ordinates x, y, z,
of each particle at a surface of displacement change to x + udt, y + vdt,
z + wdt, and are ultimately the co-ordinates of the surface of displace-
ment in a new position indefinitely near the former. Hence

<p (x + udt, y + vdt, z + wdt, t + dt) = 0,

Q Q2

352 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE

or, putting (p for <p (s, y, as, t),

<t> + -*£ . udt + -S- . vdt + ~- . wdt + -T7 dt = 0.
r dx dy dz dt

Now = 0, -j^ = Nu, -j- = Nv, -S = Nw. Hence
r dx dy dz

N(u* + v° + w 2 ) + S = 0.

Or, if u* + r" + lb" =V\ N = - ^- f .

Let, for example, the surface of displacement be that of a sphere of
given radius moving with a given velocity V t in the direction of the axis
of %. Then if R be the radius of the sphere, and a, b, c, be the co-
ordinates of its centre,

(*, y, %, f) = {x - a)' + (y- bf + (u - cf - R\

rr dc d(b ', . dc „ rr , .

v = d-r di = -^ % - c ^r-^ v ^-^

and the normal velocity Vis equal to V r _ . Consequently N = j^-. .

Si r t \% — C)

It therefore appears that the surface of an oscillating sphere may be a surface

of displacement, and that the factor JV varies as g » as I have supposed

in Art. 5. It also appears that the error of Poisson's solution consists in his
employing an equation depending on the supposition that udx + vdy + wdz
is of itself an exact differential ; a supposition which, as we have seen, is
not of sufficient generality. Indeed it would not be difficult to shew that
this condition is fulfilled only when the surfaces of displacement coincide
with surfaces of equal pressure during the whole of the motion, and when
in consequence the motion of each particle of the fluid is rectilinear. The
differential equations of fluid motion in their most general form have never
yet been obtained.

The above considerations lead to a very simple solution of the problem
of the resistance of the air to an oscillating sphere. For supposing the
motion of the sphere to be impressed on the sphere and on the air in the

ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 353

direction contrary to that of the sphere's motion, the sphere will be reduced
to rest, and the velocity of the fluid along the surface of the sphere, from
what is proved above, will be V t sin 0. Hence by a known Theorem of
Hydro-dynamics,

p'Bde*\ dt j _u '

and as jo = a*f>, and -jj- - V t sin 0, if we put m <p (t) for V t , it follows

that,

^^+mB<p'(t) sin 9 + \<f> (t)}* cos 6 sin = 0;

(2

whence, by integration,

tf Nap. log. P - R<p' (t) cos 9 ~ 1JL cos 2 = a function of t.

This equation agrees in its ultimate application with (10) of Art. 5,
and consequently leads to the same result.

Cambridge Observatory,
Dec. 13, 1841.

XX. Description of an Extinct Lacertian Reptile, Rhynchosaurus arti-
ceps, Owen, of which the Bones and Foot-prints characterize the
Upper New Red Sandstone at Grinsill, near Shrewsbury. By
Richard Owen, F. R. S., G. S. &f., Hunterian Professor in the
Royal College of Surgeons.

[Read April 11, 1842.]

The existence of a small four-footed animal, at the period of the depo-
sition of the New Red Sandstone near Shrewsbury, was announced by
Dr. Ogier Ward of that city, at the meeting of the British Association at
Birmingham ; the evidence then brought forward consisting of foot-prints
only. These Ichnolites most nearly resembled those figured in the Memoir
on the New Red Sandstone of Warwickshire, by Messrs. Murchison
and Strickland*, but differed in giving more distinct indications of the
terminal claws, and less distinct impressions of the connecting web : the
innermost toe is more diminutive, and there is an impression, always at
a definite distance from the fore-toes, like a hind-toe pointing backwards,
and which seems to have only touched the ground by its point, as in some
wading birds : reminding one of the form of some of the Ichnolites dis-
covered by Dr. Hitchcock, in the New Red Sandstone at Connecticut,
which have been referred to the class of birds.

Any evidence of a warm-blooded and quick-breathing class of animals
at so remote a period as the New Red Sandstone epoch requires to be
very closely sifted, and the chance of obtaining any analogical facts,
bearing upon the explanation of the * Ornithicnites' of Professor Hitchcock,
induced me to spare no exertions to obtain further insight into the
problematical creature of the Grinsill quarries.

* Geological Transactions, Second Series, Vol. V. pi. xxviii. .

356 PROFESSOR OWEN, ON THE RHYNCHOSAURUS.

Through the kind and zealous attention of Dr. Ward to the quarrying
operations in his neighbourhood, various fossils were from time to time
secured and transmitted to me, which at length enabled me to form
a clear opinion of the nature and affinities of the animal in question.
I say of the animal that impressed the sands with its feet, because, with
respect to the bones, Dr. Ward observes in his letter accompanying them :
" As they have always been found nearly in the same bed as that impressed
by the footsteps I have described, I am induced to believe that these are
the bones of the same animal:" and in this opinion, from the correspondence
of size between the bones and foot-prints, and from the circumstance
of the absence of other observed bones or foot-prints in the same quarry,
I entirely coincide.

The vertebrae, to which my attention was first directed, proved the
species to belong to the Lacertine or lower division of the great Saurian
group of reptiles*. These bones will be first described.

Vertebree. — Both surfaces of the centrum are concave and are deeper
than in the biconcave vertebrae of the extinct Crocodilians ; the texture
of the centrum is compact throughout. In the dorsal series the two
lateral surfaces join the under surface at a nearly right angle, the transverse
section presenting a subquadrate form, with the angles rounded off: the
under surface and sides are regularly concave longitudinally.

The neural arch is anchylosed with the centrum, without trace of suture,
as in most Lizards ; it immediately expands and sends outwards from

* An extended survey of the modifications of this class of Vertebrata from their first appear-
ance on the Earth's surface to the present time, is necessarily attended with different views of
their classification than can be derived from an acquaintance, however close, with existing
species only. I propose to divide the Reptilia into eight orders: viz. Dinosauria, Enalio-
sauria, Crocodilia, Lacertili a, Pterosauria, Chelonia, OPHiDiA.and Batrachia. They
are here enumerated in the descending scale of organization. The Saurian division was repre-
sented of old by reptiles manifesting the crocodilian grade of structure, under a rich variety of
modifications, constituting, besides the typical and still represented groups, two other orders,
now wholly extinct ; it has since subsided into a swarm of small Lacertians, headed by so few
examples of the Crocodilian or Loricate species, that it is no marvel such relics of a once pre-
dominating tribe should have found a humble place in Linne's System of Nature, as co-ordinate
members of the genus Lacerta.

PROFESSOR OWEN, ON THE RHYNCHOSAURUS. 357

each angle of its base a broad triangular process with a flat articular
surface; the two anterior surfaces look directly upwards, the posterior
ones downwards ; the latter are continued backwards beyond the posterior
extremity of the centrum; the tubercle for the simple articulation of
the rib is situated immediately beneath the anterior oblique process. So
far the vertebra? of the Rhynchosaurun, always excepting their biconcave
structure, resemble the vertebra of most recent lizards. In the modifica-
tion next to be noticed, they show one of the A*ertebral characters of the
Dinosauria*. A broad obtuse ridge rises from the upper convex surface
of the posterior articular process, and arches forwards along the neur-
apophysisf above the anterior articular process, and gradually subsides an-
terior to its base: the upper part of this arched angular ridge forms, with
that of the opposite side, a platform, from the middle line of which the
spinous process is developed. This structure is not present in existing
lizards; the sides of the neural arch in their vertebras immediately
converge from the articular processes to the base of the spine, without
the intervention of an angular ridge formed by the side of a raised
platform. The base of the spinous process in the Rhynchosaur is broadest
behind, and commences there by two roots or ridges, one from the upper
and back part of each posterior articular process : they meet at the
posterior part of the summit of the neural arch, whence the spinous
process is continued upwards as a simple plate of bone, its base extend-
ing forwards along about two thirds of the length of the platform,
which then again divides into two ridges which diverge from each other
in slight curves to the anterior and external angles of the neurapo
physes. The interspace of the diverging anterior crura of the base of
the spine is occupied by a triangular fossa, not continued into the sub-
stance of the spine; this fossa is bounded below by a horizontal plate
of bone extended over the anterior part of the spinal canal, and ter-
minated by a convex outline. The anterior margin of the spinous

* The characters of this extinct Order of Reptiles are given in the Report of the British
Association, 1841, p. 102.

t The vertebral nomenclature, which I have been compelled to invent -for the requisite
clearness and brevity of description of these most complicated and most common of Reptilian
fossils, is explained in the Geological Transactions, Vol. V, pt. iii, Second Series, p. 518.
Vol. VII. Part III. Rg

358 PROFESSOR OWEN, ON THE RHYNCHOSAURUS.

process is thin and trenchant ; the height of the spine does not exceed
the antero-posterior diameter of its base ; it is obliquely rounded off. The
spinal canal sinks into the middle part of the centrum, and rises to
the base of the spine, so that its vertical diameter is twice as great at
the middle as at the two extremities : this modification resembles, in
a certain degree, that of the vertebra? of the Palceosmirus from the
Bristol conglomerate*. The following are dimensions of the most perfect
of the dorsal vertebra? of the Rhynchosaurus : —

Lines.

The length of the centrum 5-£-

Height of the articular end 3

Breadth of the articular end 2-f- ^

From the lower margin of the posterior extremity of the cen-
trum to the posterior part of the base of the spine 5

From the lower margin of the posterior extremity of the cen-
trum to the summit of the spine 9

Antero-posterior extent of base of spine 4

Breadth of the neural arch, from the outer margin of one

anterior articular process to that of the opposite side 8-J-

Breadth of the neural arch at the interspace between the ante-
rior and posterior articular processes 4

Breadth of the neural arch across the middle of the spinous
platform 2

Skull. — The most complete specimen yet obtained of this instructive
part of the skeleton of the Rhynchomurus is imbedded in a portion of
the coarse-grained sandstone from the Grinsill quarries. The lower jaw
is in its natural position, as when the mouth is shut, showing that the
parts had not been dislocated when they became imbedded in the sand.

The skull presents the form of a four-sided pyramid, compressed
laterally, and with the upper facet arching down in a graceful curve to
the apex, which is formed by the termination of the muzzle.

The very narrow cranium, the wide temporal fossa on each side,
bounded behind by the bifurcations of the parietal and the mastoid, and
laterally by a strong compressed zygoma, with a long tympanic pedicle
descending vertically from the point of union of the transverse and

* Geological Transactions, Second Series, Vol. V. p. 349, pi. xxix.

PROFESSOR OWEN, ON THE RHYNCHOSAURUS. 359

zygomatic arches, and terminating in a convex pulley for the articular
concavity of the lower jaw, — the large and complete orbits, and the
short, compressed, and bent-down maxilla?, — all combine to prove the
fossil to belong to the Lacertine division of the Saurian Order.

The lateral compression and depth of the skull, the great vertical
extent of the superior maxillary bone, the small relative size of the
temporal spaces, the great depth of the lower jaw, prove that it does
not belong to a reptile of the Batrachian Order. The shortness of the
muzzle, and its compressed form, equally remove it from the Crocodilians.
No Chelonian has the tympanic pedicle so long, so narrow, or so freely
suspended to the posterior and lateral angles of the cranium.

The general aspect of the skull differs, indeed, from that of existing
Lacertians, and singularly resembles that of the bird or turtle, and the
resemblance is increased by the apparent absence of teeth. The inter-
maxillary bones, moreover, are double, as in the Chelonia, and also
symmetrical, not united by a median ascending process ; but, with this
exception, all the more essential characters of the skull are those of
the Lizard.

Of the proper parietes of the cerebral cavity, the portion formed
by the parietal and frontal bones is exposed. The parietal is traversed
by a thin, but high median crest longitudinally : the sides are convex,
and the breadth of the bone diminishes towards the occiput : here it
divides into two branches, which pass outwards, more transversely than
in existing Lizards. There is no perforation either in the parietal
bone, or in the coronal suture. This suture is transverse. At the an-
terior part of the parietal crest two lines diverge from each other at
a right angle to the upper part of the orbit, and separate the median
from the post-frontals ; a nearly transverse suture divides the fore-part
of the parietal from the post-frontals. The median frontal bone is single,
like that of the new-world Thorictes {Thorictes, Tejus) and Iguanas, not
divided, as in the Varanians. It expands slightly as it advances towards
the fore-part of the orbits, the oblique lines dividing the median frontal
from the post-frontals, and the supra-orbitary ridges are raised, so that
the interspace is slightly concave ; and the surface is also broken by

BBS

360 PROFESSOR OWEN, ON THE RHYNCHOSAURUS.

irregular elevations and depressions. The post-frontal is divided by a
nearly transverse suture. This bone completes the upper and outer part
of the orbit by a thin well-defined curved plate; an irregular blunt
ridge descends in a nearly vertical direction behind this plate, and
then the posterior frontal is continued backwards in the form of a long
compressed plate gradually terminating in a point, which overlaps the
zygomatic bone. This forms the medium of union between the long
posterior frontal and the parietal fork. The posterior frontal is divided
by a suture, as in the Iguana?; the anterior division forms a projecting
curved plate at the upper and outer part of the circumference of the
orbit, and prolongs forward the rim of that cavity beyond the plane of
the head.

The tympanic bone is bent like the Italic f, and is slightly expanded
transversely at its distal extremity : its posterior surface is exposed in
the present fossil, showing it to be convex and rounded, and continued
externally in the form of a thin plate, which is concave posteriorly.
The thick convex stem divides near the lower end into two ridges,
which diverge, like the condyles of a humerus, and intercept the trochlea
on which the concave articulation of the lower jaw plays. The tympanic
trochlea is convex from behind forwards, concave from side to side.

The orbit is large, nearly circular in form, and its bony frame is
complete; this is formed above by the median, anterior and posterior
frontals, before by the anterior frontal and lachrymal, below by the
malar, and behind by the malar and posterior frontal.

The malar bone, as in most lizards, is long, slender, and bent upon
itself, but its external surface is unusually concave, the orbital plate
projecting outwards, like the corresponding rim formed by the frontal
bone. The anterior or horizontal branch of the malar gradually tapers to
a point, which is wedged in between the lachrymal and superior max-
illary ; the posterior branch ascends at nearly a right angle and is
applied obliquely to the posterior part of the descending process of
the posterior frontal : at the angle between the two portions of the
malar a process is continued backwards for about half an inch, but its
extremity is broken off.

PROFESSOR OWEN, ON THE RHYNCHOSAURUS. 361

The lachrymal bone presents the same relative size and position as
in the Thorictes, Lacerta, and most Lizards; a tubercle rises from
about the middle of its external surface.

The superior maxillary is a broad vertical triangular plate of bone,
with a smooth external surface : the alveolar border projects externally
like a ridge, above which the bone is slightly concave : this ridge
appears to be slightly dentated, and overlaps the corresponding alveolar
border of the lower jaw. The posterior superior margin of the max-
illary bone is slightly concave, and joins the malar and lachrymal bones,
and a small part of the prefrontal : the anterior superior margin joins
the upper half of the elongated intermaxillary bone, which divides it from
the nasal bone and the external nostril: the lower side or base of
the triangle, which forms the alveolar border, is convex.

The most singular character of the cranium of the present fossil
Reptile is afforded by the intermaxillary bones. These, in their length
and regular downward curvature, give to the fore-part of the skull
the physiognomy of that of an accipitrine bird; but they differ es-
sentially from both those of the bird and lizard, in being on each
side distinct throughout their whole length, and in gradually diminish-
ing to their inferior or rostral extremity, which is not expanded or
continued laterally to form any part of the alveolar border of the
upper jaw. Each intermaxillary bone is a slender subcompressed elon-
gated bone, bent so as to describe a quarter of a circle ; the upper
half is thinner, but rather broader, than the lower one, and is wedged
in between the superior maxillary, frontal, and nasal bones: the lower
half, which is somewhat narrower, but thicker and subcylindrical, pro-
jects freely downwards beyond the superior maxillary hone ; and the deep
anterior extremity of the lower jaw is applied to the posterior surface
of these produced extremities of the two intermaxillaries when the
mouth is closed. The two intermaxillaries converge towards each other
from their posterior origins, and are in close contact with each other
where they form the singular curved and prominent beak.

The external nostril I presume to be situated between the upper
diverging ends of the intermaxillaries, but a fracture of the fossil at

362 PROFESSOR OWEN, ON THE RHYNCHOSAURUS.

this part prevents the determination of the precise form of this aperture,
or the mode of termination of the nasal bones. These bones, if not
actually absent in the present fossil, as in most Chelonia, must have
been extremely small, as in the Chameleon.

The lower jaw is of considerable depth, and exceeds, as in most
Saurians, the length of the cranium. The articular cavity is deep and
wide: the angle of the jaw is broken off in the fossil directly behind
this cavity on the left side, but is continued backwards beyond it
for more than half an inch on the right side. The ramus gradually
expands in the vertical direction, and becomes thinner from side to
side, as it advances forwards, to about its middle part, which is just
behind the orbit, where it measures 11 lines in depth : it then be-
gins gradually to diminish vertically to the symphysis, which again
slightly increases to its termination, which is obliquely truncated, much
compressed laterally, and applied against the deflected extremities of
the intermaxillaries. The posterior half of the maxillary ramus is slightly
convex externally ; the anterior narrower part slightly concave : the
superior margin describes a slight but graceful sigmoid curve, convex
posteriorly, and concave anteriorly, where it is adapted to the convex
alveolar border of the upper maxillary bone, to the inner side of which
it is closely applied. The alveolar border forms an external, convex,
projecting ridge, analogous to that of the upper jaw.

The composite structure of the lower jaw is very clearly displayed
in the fossil. The articular piece is short, but is continued forwards
as a slender process below the angular piece e, as in the Varanus. The an-
gular piece is relatively larger than in Varanus, and presents nearly
the same proportions as in Thorictes. The supra-angular is longer, and
occupies the proportion of the jaw formed by the supra-angular and
coronoid elements in Thorictes and other Lizards. The opercular element
extends farther upon the outside of the jaw from its lower margin
than in the existing Lizards; Thorictes, again, in this respect, coming
nearest to Rhynchosaurus. The dentary element constitutes the rest of
the outer side of the ramus, but not the slightest trace of teeth is
discernible. The fossil seems to have been preserved with the mouth

PROFESSOR OWEN, ON THE RHYNCHOSAURUS. 363

naturally closed, and the upper and lower jaws are in close contact.
In this state they must originally have been buried in the sandy
matrix, which afterwards hardened around them : and since true Lizards,
owing to the uninterrupted succession of their teeth, do not become
edentulous by age, we must conclude that the state in which the
Rhynchosaurus was buried, with its lower jaw in undisturbed articu-
lation with the head, accorded with its natural state while living, so
far as the less perishable parts of its masticatory organs were con-
cerned. Nevertheless, since a view of the inner side of the alveolar
border has not been obtained, we cannot be ass