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 necessary 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 for 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 intensities 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-velocities, and in direction, the directions of the three corresponding disturbances by which a wave can propagate itself in one 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 direc

tions 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 t.

Let us conceive a system composed of an immense number of particles 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 dr, dy, dz in the primitive state of the element. Then will

2

(dy) } dy'

(de')2 = {(dez)
{(d')" + ('db')"' + (de)"} de' = c'da'
(税)+(税)

* [It will be seen that this remark is not strictly correct, as the surface must necessarily have another common plane section.]

+ [Referring to the values of u, v, w given in p. 301, we see that, since the direction of vibration is supposed to be in the front of the wave, we have

Suppose now, as in a former paper, that pdx dydz is the function due to the mutual actions of the particles which compose the element whose primitive volume = dx dy dz. Since p must remain the same, when the sides (d'), (dy'), (dz') and the cosines a, ß, y of the angles of the elementary oblique-angled parallelopiped remain unchanged, its most general form must be p=function (a, b, c, a, ß, y),

or since a, b, and c are necessarily positive, also

a=bca, Bacß, and y=aby,

This expression is the equivalent of the one immediately preceding, and is here adopted for the sake of introducing greater symmetry into our formulæ.

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 sym

metry with respect to the plane (yz), will require to remain unchanged, when we change

But thus a2, b2, c2 and a' evidently remain unaltered; moreover

Applying the like reasoning to the other co-ordinate planes, we see that the ultimate result will be

&=ƒ (a3, b2, c2, a'2, B12, y'2)

.(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 + v,

z = z +20,

u, v, and w being very small functions of x, y, z, and t of the first order. Then by substitution we get

we thus see that 8, §,, § ̧, d'‚ ß', y', are very small quantities of the first order, and that the general formula (1) by substituting the preceding values would take the form

which

$ = function (§,, 8,, dg, d', B', I),

may be expanded in a very convergent series of the form

$= &%+ &¿ + &2+ &,+&c. :

P., P., 4., &c. being homogeneous functions of 8, &, &q Ú, B., Ÿ‚ of the degrees 0, 1, 2, 3, &c. each of which is very great compared with the next following one.

But being constant, if p the primitive density of the eleinent, the general formula of Dynamics will give

(d'u d'v
de

δυ

[[{pdxdyda {du bu+ de 8v+ Bio} = [[[dzdydz(&4,+84,+&c).

de

If there were no extraneous pressures, the supposition that the primitive state was one of equilibrium would require 1 = 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 4,, ., &c. will be insensible and the preceding formula may be written

Supposing p the primitive density constant, the most general form of 4, will be

Þ1 = − — (.As ̧ + B§ ̧ + C§ ̧+2Da' + 2ES' +2Fy'),

Φι

A, B, C, D, E, and being constant quantities.

In like manner the most general form of 4, will contain twenty-one coefficients. But if we first employ the more parti