but the form of 4 so determined is different from Cauchy's, and leads to a more complicated solution. On the whole I cannot see that Cauchy's theory of reflection has any claim to be considered dynamical, although his formulæ are, beyond doubt, very good empirical representations of the facts.

I now come to the modification of Green's theory proposed by Haughton. If M were an arbitrary constant instead of a definite function of μ, there would be but little difference between the two sets of formulæ; for the factor sin would not vary greatly in the neighbourhood of the polarizing angle, where alone the correction to Fresnel's original expression is sensible. So far as the question has been treated experimentally, the balance of evidence seems to be rather against than for the factor sin . I have already reinarked that Haughton's reasons for considering M as an independent constant cannot be sustained, but at the same time I think that others of considerable force may be given.

In a supplement to his memoir "On the Reflection of Light"*, Green says:-"Should the radius of the sphere of sensible action of the molecular forces bear any finite ratio to A, 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 ætherial medium in the immediate vicinity of their surface of separation. And I have here endeavoured to show by probable reasoning that the effect of such a change would be to diminish greatly the quantity of light reflected at the polarizing angle, even for highly refractive substances, supposing the light polarized perpendicular to the plane of incidence." The contrast between this view and that of Lorenz is remarkable.

Referring to equation (9), we see that when =n, it reduces to

m(a'2+b2)$=m' (a¦2+b2) $,•

Reasoning from the analogy of elastic solids, we found

m(a12 + b2) : m' (a¦2 + b2)=D: D'. . (11)

Now although the transition between the two media is so sudden that the principal waves of transverse vibrations are affected nearly in the same way as if it were instantaneous, yet we may readily imagine that the case is different for the surface-waves, whose existence is almost confined to the layer of variable density. It is probable that the ratio of m(a12 +b2) : m'(a¦2+b2), instead of being equal to 1: μ2, approaches much more nearly to

* Cambridge Trans. 1839, or Green's works.

This explanation of the deviation of M from Green's value seems to me the most probable; but the ground might be taken that the densities concerned in the propagation of the so-called longitudinal waves are unknown, and may possibly not be the same as those on which transverse vibrations depend. For sulphuret of arsenic, Jamin's experiments give

μ=2·454, μ=1.083,

showing that po is very considerably less than μ.

One of the most remarkable of Jamin's results shows that in many cases M is negative, or μo less than unity. There are a few substances of an intermediate character for which M=0; and then Fresnel's original formulæ express the laws of the phenomena. The value of is usually about 1:45. No adequate explanation has hitherto been given of the singular law; and in the remarks which follow I wish to be understood as merely throwing out a suggestion which may or may not contain the germ of an explanation.

μ

It is known that many solid bodies have the power of condensing gases on their surfaces, a property on which the action of Grove's gas-battery seems to depend. Now, if we were to suppose that at the surfaces of solid and liquid bodies there exists a sheet of condensed air, which need not extend to a distance greater than the wave-length, but is of an optical density corresponding to about μ=1.5, the occurrence of negative values of M would, I think, be explained. There is nothing à priori very improbable in the existence of such a sheet, so far as I am able to see; but it is for experiment to decide whether the phenomena observed near the polarizing angle depend in any manner on the nature of the gas with which the reflecting body is in contact, and whether the sign of M may change from negative to positive when vacuum is substituted for atmospheric air. The fact that the value of M for the surface of separation of (say) glass and water cannot be calculated from the values of M corresponding respectively to glass and air, water and air, seems to indicate that the phenomenon is, so to speak, of an accidental character.

Phil. Mag. S. 4. Vol. 42. No. 278. Aug. 1871.

H

XII. On Mr. Hopkins's Method of determining the Thickness of the Earth's Crust. By Archdeacon PRATT, M.A., F.R.S.

To the Editors of the Philosophical Magazine and Journal.
GENTLEMEN,

IN my

my letter in your Number for July of last year I gave a reply to M. Delaunay on the above subject in a popular form, and in that way explained that, even if the fluid interior of the earth's mass at any moment revolved with the crust as if the two were one solid mass, this state could not continue; the crust under the action of the precessional force would slip over the fluid, not being solidly connected with it.

M. Delaunay (as reported in 'Nature,' March 16, 1871, column 1) has again said that "calculations prove " that the

thickness of the crust has "no influence on the revolution of the earth." I therefore now send you a calculation to show that the crust, with an interior fluid nucleus, both following the law of density adopted by Laplace, cannot move as it would if the crust and nucleus were one solid mass. The method I pursue is this. At the epoch from which t (the time) is measured I assume that things are exactly as M. Delaunay supposes, viz. that internal friction and viscosity have reduced the fluid to entire obedience at that moment to the movements of the crust. I then show by the equations that this state of things cannot continue. This mode of taking the problem enables me easily to calculate the effect of the fluid pressure on the crust at the epoch; and any minute motion, gradually generated in the fluid after this for a short time, would enter into the equations as a quantity of the second and higher orders and may be neglected. The slowness of the disturbed motion and the viscosity would have the effect of prolonging the period through which my equations would apply.

2. The forces acting on the crust are the attraction of the sun and moon from without, and the pressure of the fluid against its interior surface; and the pressure of the fluid is produced by the centrifugal force and the attraction of the sun and moon on the fluid. As the crust is supposed to be made up of spheroidal shells, it will have no effect on the fluid.

Let w1, w, wg be the angular velocities of the solid crust round any line at right angles to the earth's axis, another axis at right angles to it and lying in the plane of the equator, and the axis of rotation; i. e. the axes of x, y, z fixed in the body; A, A, C the moments of inertia about those three axes; L, M the moments of the forces acting on the crust about the axes of a and y. There will be no moment of forces about z, because the resultant effect of each set of forces passes through the axis

of z, owing to the symmetry of figure. Hence

The third equation gives wa, constant =n; and the others become

3. The values of L and M for the sun and moon are known from the ordinary problem of precession and nutation; they are easily shown to be

(C— A)

3S
C-3

3S
C

sin cos 0 sin o, -(C-A) sin e cose cos

for the sun, where S is the sun's mass, c his mean distance, and his colatitude and right ascensiont. Similar expressions are true for the moon's action. Let them be

(C-A) sin e, cos e, sin,, -(C-A) 0,

3M
c

3M c,8

sin e, cos e,coso,

4. I will now find L and M for the pressure of the fluid. Let r, e', ' be the coordinates to any point in the inner surface of the crust, and p the fluid pressure. Then pr2 sin e' do'de' is the pressure on an element of the surface, and acts in the normal. Let be the angle the normal makes with the earth's axis. Then by conics

1=0'-2e cos 0' sin e',

e being the ellipticity of the inner surface of the crust. Hence the pressures parallel to the axes are

and

also

pr2 sin O'dp'de'. sin l cos p', pr2 sin O'do'de'. sin l sin p',

pr2 sin 'do'de'. cos l;

x=r sin e' cos p', y=r sin e' sin p', z=r cos 0'.

*These equations will be found in the Mécanique Celeste, or any work on the motion of a rigid body. I have taken them from my 'Mechanical Philosophy,' second edition, p. 425, making B=A.

† Mechanical Philosophy, p. 426.

Hence the moment of the fluid pressure on this element about the axis of x

=pr3 sin 0'do'd0' (cos l. sin e' sin p' — sin l sin d'. cos 0)

=pr3 sin 0'dp'de' sin p' cos (l—0')=2epa3 sin20' cos 0'sin p'de'dp', putting ra, the mean radius, because the square of e may be neglected. Integrating for the whole surface, putting cos '=', the part of L which depends on the fluid pressure

Similarly the part of the moment M which depends on fluid pressure

The function under the signs of integration is a Laplace's function of the second order.

5. I must now find p. The centrifugal forces on any particle (a'y') of the fluid parallel to x and y are n2x' and n2y'. Also, if R is the distance of the sun from that particle, the attraction of the sun on it

d S

=- dR R'

and similarly of the moon; and the equation of fluid equilibrium at the epoch gives

r', ', ' being the coordinates to the particle of fluid. I shall at present leave out M (the moon), and, when the effect of S is found, add a similar term for M. Now

P1, P2,... being Laplace's coefficients. I shall integrate dp from the earth's centre, along r', to the surface of the crust, keeping ' and ' constant. Then

S

H') { 'p'r'dr' + 3 (P1 ('p'dr + 2 P. ("pr'dr'+.).

μ 3

Spar

Substitute this in the formulæ of the last paragraph, observing that as e2 is to be neglected, the means a and a' may be put for r and . Observing the properties of Laplace's functions, and