speak, not of the absolute displacement within a refracting medium, but of the equivalent displacement in vacuum, of which all that we are concerned to know is, that it is proportional to the absolute displacement. By the equivalent displacement in vacuum, is here meant the displacement which would exist if the light were to pass perpendicularly, and therefore without refraction, out of the medium into vacuum, without losing vis viva by reflection at the surface. It is easy to prove that Fresnel's formula for refraction would be adapted to this mode of estimating the vibrations by multiplying by μ; indeed, the formulæ for refraction might be thus proved, except as to sign, by means of the principle of vis viva, the formulæ for reflection being assumed. It will be sufficient to shew this in the case of light polarized in the plane of incidence.

Let i, be the angles of incidence and refraction, A any area taken in the front of an incident wave, the height of a prism having A for its base and situated in the first medium. Let r be the coefficient of vibration in the reflected wave, that in the incident wave being unity, q the coefficient of the vibration in vacuum equivalent to the refracted vibration. Then the incident light which fills the volume Al will give rise to a quantity of reflected light filling an equal volume Al, and to a quantity of refracted light which, after passing into vacuum in the way supposed, would fill a volume Al cos i/cos i. We have therefore, by the principle of vis viva,

This equation does not determine the sign of q: but it seems impossible that the vibrations due to the incident light in the ether immediately outside the refracting surface should give rise to vibrations in the opposite direction in the ether immediately inside the surface, so that we may assume q to be positive. We

as was to be proved. The formula for light polarized perpendicularly to the plane of incidence may be obtained in the same way. The formula (5), as might have been foreseen, applies equally well to the hypothesis that the diminished velocity of propagation within refracting media is due to an increase of density of the

ether, which requires us to suppose that the vibrations of plane polarized light are perpendicular to the plane of polarization, and to the hypothesis that the diminution of the velocity of propagation is due to a diminution of elasticity, which requires us to suppose the vibrations to be in the plane of polarization.

If the refraction, instead of taking place out of vacuum into a medium, takes place out of one medium into another, it is easy to shew that we have only got to multiply by / instead of √μ; μ, μ being the refractive indices of the first and second media respectively.

[From the Cambridge and Dublin Mathematical Journal, Vol. iv. p. 194

(May and November, 1849).]

ON ATTRACTIONS, AND ON CLAIRAUT'S THEOREM.

CLAIRAUT'S Theorem is usually deduced as a consequence of the hypothesis of the original fluidity of the earth, and the near agreement between the numerical values of the earth's ellipticity, deduced independently from measures of arcs of the meridian and from pendulum experiments, is generally considered as a strong confirmation of the hypothesis. Although this theorem is usually studied in connection with the hypothesis just mentioned, it ought to be observed that Laplace, without making any assumption respecting the constitution of the earth, except that it consists of nearly spherical strata of equal density, and that its surface may be regarded as covered by a fluid, has established a connexion between the form of the surface and the variation of gravity, which in the particular case of an oblate spheroid gives directly Clairaut's Theorem*. If, however, we merely assume, as a matter of observation, that the earth's surface is a surface of equilibrium, (the trifling irregularities of the surface being neglected), that is to say that it is perpendicular to the direction of gravity, then, independently of any particular hypothesis respecting the state of the interior, or any theory but that of universal gravitation, there exists a necessary connexion between the form of the surface and the variation of gravity along it, so that the one being given the other follows. In the particular case in which the surface is an

* See the Mécanique Céleste, Liv. III., or the reference to it in Pratt's Mechanics, Chap. Figure of the Earth.

oblate spheroid of small eccentricity, which the measures of arcs shew to be at least very approximately the form of the earth's surface, the variation of gravity is expressed by the equation which is arrived at on the hypothesis of original fluidity. I am at present engaged in preparing a paper on this subject for the Cambridge Philosophical Society: the object of the following pages is to give a demonstration of Clairaut's Theorem, different from the one there employed, which will not require a knowledge of the properties of the functions usually known by the name of Laplace's Functions. It will be convenient to commence with the demonstration of a few known theorems relating to attractions, the law of attraction being that of the inverse square of the distance*.

Preliminary Propositions respecting Attractions.

PROP. I. To express the components of the attraction of any mass in three rectangular directions by means of a single function.

Let m' be the mass of an attracting particle situated at the point P', the unit of mass being taken as is usual in central forces, m the mass of the attracted particle situated at the point P, x, y, z' the rectangular co-ordinates of P′ referred to any origin, x, y, z those of P; X, Y, Z the components of the attraction of m' on m, measured as accelerating forces, and considered positive when they tend to increase x, y, z; then, if PP' =r',

*

My object in giving these demonstrations is simply to enable a reader who may not have attended particularly to the theory of attractions to follow with facility the demonstration here given of Clairaut's Theorem. In speaking of the theorems as "known" I have, I hope, sufficiently disclaimed any pretence at originality. In fact, not one of the "propositions respecting attractions" is new, although now and then the demonstrations may differ from what have hitherto been given. With one or two exceptions, these propositions will all be found in a paper by Gauss, of which a translation is published in the third volume of Taylor's Scientific Memoirs, p. 153. The demonstration here given of Prop. iv. is the same as Gauss's; that of Prop. v., though less elegant than Gauss's, appears to me more natural. The ideas on which it depends render it closely allied to a paper by Professor Thomson, in the third volume of this Journal (Old Series), p. 71. Prop. IX. is given merely for the sake of exemplifying the application of the same mode of proof to a theorem of Gauss's.

with similar equations for Y and Z.

If instead of a single particle m' we have any number of attracting particles m', m"... situated at the points (x', y', z'), (x", y', z'')..., and if we put

If instead of a set of distinct particles we have a continuous attracting mass M', and if we denote by dm' a differential element of M', and replace (1) by

equations (2) will still remain true, provided at least P be external to M'; for it is only in that case that we are at liberty to consider the continuous mass as the limit of a set of particles which are all situated at finite distances from P. It must be observed that should M' occupy a closed shell, within the inner surface of which P is situated, P must be considered as external to the mass M'. Nevertheless, even when P lies within M', or at its surface, the

expressions for V and dV/dæ, namely S

dm'

and

(x'

dm' 23

admit of real integration, defined as a limiting summation, as may be seen at once on referring M' to polar co-ordinates originating at P; so that the equations (2) still remain true.

PROP. II. To express the attraction resolved along any line by means of the function V.

Lets be the length of the given line measured from a fixed point up to the point P; λ, μ, v, the direction-cosines of the tangent to this line at P, F the attraction resolved along this tangent; then

F =λX+μY+vZ= λ·

dv dV
+v

dv
dx

dz