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We will change the variables in this expression by writing cos 0 =p and r=ah. The expression then becomes

h (1-hu) ap

dh du do,

(1-2pk + ?) where we have rejected the negative sign that arises in the transformation from 0 to M, as we afterwards intend to integrate with regard to u in the direction in which that variable increases.

Now, if h be less than unity, as it is in the case we are considering, we may write

1

=1+P+P,+ ... +h"P,+&c., (1-2 Al +12 ); and therefore (1 – 24.11 + 1)2 =- +] (uP,-1) + ... +h"(uPn - P.,-1)+&c. Now, we have the formula

(2n +1) Pn=(n+1)Pn+1+nP,-1, and therefore

n+1
uP-Pn-1= (Pn+1-Pn-1).

21+1 Thus we obtain

Mh

n+1

M(1–2uh+hi) 3 = x +şl(P,-1) + ... +

hon (PM+1-P.-1) + &c.

21+1

dP,

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1+31(P)

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+ &c.

dP,+1

If we differentiate this with respect to M, we deduce 1-uh

n+1

dP,

'n+1 +

le"

+ (1–2uh + ha)!

du
2n + 1

du du
=1+2hPi + ... + (n +1) "P+ &c.,
since

=(2n +1)P, du Thus the expression for the portion contributed to the value of gravity at P by our element is

ap dh du d${I2 + 2HP,+ ... + (n-1))"Pn-3+&c.}. Imagine a solid element cut out of the crust by radii drawn from the Earth's centre to all points of the contour of a polar surface-element drawn upon the Earth's surface. The portion contributed to the value of gravity at P by our solid element

dP,-1

du

can be obtained from the above expression by direct integration. We will not write down the whole series, but only the term obtained by the integration of that containing li". The other terms can be deduced from this. The value of this term will be

1 a du do

n+1 where

=

{1-(1-)**}+{(14)*** - (1-3)*** } p.

+{{1- ) *** - (1-4)** }

+...+

We will now introduce the following assumptions :

(e-pak+ ... + (pm-1-pm)km-1 + pmkm=aA,,
(pa-pa) ki + +(em-1-pm) k_1 +Pmken=a’A,,

(P1-P2)kļ + ... +(m-1-pm) hein-, +Phon=aPA,, where A1, A2, ..., A, are constant throughout the entire extent of the crust. The first (p—2) terms of our series expressing the portion contributed to the value of gravity at P by our solid element, may then be written

a du {Ci+2C,P,+ +(p-2)Cp-2P0-3}, where

Ci=A1-A, + $A3,
Cy=A,-A, +A3-444,

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The (p-1)th term of the series may be written in the form

a du do(p-1)P0-2(Co-1+ U), where

p(p-1) C,-1=A-”, Ag+

- ...+(-1)p-1Ap) ,

1.2.3

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Similarly we can obtain expressions for the remaining terms.
Now

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if n be a positive integer. Hence that portion contributed
to the value of gravity by the crust, which does not depend
on the position of the point P, will be 4taC,. Besides this
there will be a minute residual effect, the most important
term in whose expression will be

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We cannot evaluate this term until we know the manner in which u depends upon u and $. As u and $ vary the quantity u will experience discontinuities in its variation, but it is doubtful if anything of the nature of an abrupt discontinuity would arise. If such discontinuities arise they must be rare. Of course at the seaboard there is an abrupt variation in the density of the topmost layer, but the depth of the sea is at first quite small as regards our problem, and afterwards increases with a fair approximation to continuity.

We have supposed m layers to exist, but this is to be taken as the maximum number. There may be less than m in certain portions of the crust, and this may be provided for by considering certain pairs of consecutive k's to be equal throughout the said portions.

The equations introduced above as assumptions are practically the same as those discussed by Mr. Fisher. It is evident that we cannot introduce more than a certain number of these assumptions without making the crust to consist of uniform concentric shells. But, working on the assumption that this is not the case, we find that we have to stay our approximation at a still earlier point than this consideration

would appear to indicate. This will be evident from a consideration of the method used on page 211 of the ‘ Physics of the Earth’s Crust,' and its subsequent applications in chapters xvii. and xxvi. We have, in each of the cases discussed, gone to the furthest approximation allowable under the circumstances. The limitations are made perfectly clear in the book, and I do not think that there is anything to be added to this part of the work.

Finally, we see that we are left with a residual effect, which is undoubtedly very small, and it is not impossible that such an effect might exist in the case of the Earth.

,

V. The Significance of Wiener's Localization of the Photo

graphic Action of Stationary Light-Waves. By J. LARMOR, F.R.S., Fellow of St. John's College, Cumbridge *. VHE

when stationary plane-polarized optical undulations are produced in a photographic film, by reflexion of a stream of incident plane-polarized light at a metallic or other backing, the photographic action occurs at the antinodes of Fresnel's vibration-vector and not at the nodes, have been employed by its author and others to decide between the various theories of light. If for purposes of precise description we utilize the terminology of the electric theory of light, which formally includes all the other theories by proper choice of the vibration-vector, we may say that the pivotographic action takes place at the antinodes of the electric vector which corresponds to Fresnel's vibration, and not at the intermediate antinodes of the magnetic vector which corresponds to MacCullagh's and to Neumann's vibration.

The crucial experiment of Wiener relates to the case when the angle of incidence is half a right angle, so that the direct and reflected waves which interfere are at right angles to each other. If the vibration take place along the direction of intersection of the two wave-planes, it will present a series of nodes and antinodes ; but if in the perpendicular direction there will not be such alternations of intensity. The experiment showed that when the light is polarized in the plane of incidence, the photographic plate develops a series of bands but when it is polarized in the perpendicular plane these bands are absent.

The argument employed is that the photographic effect will * Communicated by the Physical Society: read Norenber 9, 1894.

† Wiedemann's Annalen, 1890. Phil. Mag. S 5. Vol. 39, No. 231. Jun. 1895. H

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be greatest at those places in the stationary wave-train where the vibration is most intense ; and the conclusion is drawn from it that the actual vibration is represented by Fresnel's vector and not by MacCullagh’s ; in other words, that the vibrations of polarized light are at right angles to the plane of polarization. The force of this argument, as against MacCullagh’s theory, would, however, be evaded if the vector of that theory were taken to represent something different from the linear displacement of the æther, or if vibrations were excited in the molecule by rotation instead of translation, or by stress, as Poincaré has pointed out *.

But as a matter of fact it seems difficult to assign any reason of the above simple kind, on either theory, in favour of the photographic disturbance occurring at the antinodes rather than at the nodes of the optical vibration. The remarkable suggestion thrown out by Lord Rayleigh some time previous to Wiener's experiments, and afterwards verified by Lippmann, that certain effects in colour photography produced by Fox Talbot and Becquerel were really due to this kind of localization of the photographic effect, is not in opposition to such a view ; for the consideration adduced was simply that a localization, periodic with the waves, would, if it happened to exist, produce effects like the observed ones. At any rate, the observed localization demonstrates the important result that the effect is due to a specific dynamical action of the waves, and not to mere general absorption of the radiation.

Let us consider the actual circumstances of the case. There are about 103 molecules of the sensitive medium in the length of a single wave of light : thus in the stationary wavetrain all the parts of a single molecule would at any instant be moving with a sensibly uniform velocity, which increases and diminishes periodically. The vibration of the molecule would thus be, were it not for the influence of differences of inertia or elasticity between its parts and the surrounding æther, very nearly a swaying to and fro of it as a whole : if it were exactly this, it could not be expected to produce any breaking up of the molecule at all. Moreover, as at the antinodes of the vibration there is movement but no stress in the medium, so at the nodes there is stress but no movement; and it does not seem at all clear that alternating stress might not be as potent a factor in disintegration as alternating motion. A representation has been constructed by Lord Kelvin † of a system in which internal vibrations can be

* See a discussion on this subject in Comptes Rendus, cxii. 1891, in which MM. Cornu, Poincaré, and Potier took part.

† "Lectures on Molecular Dynamics," Baltimore, 1884.

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