equations which might very easily be proved directly in a more geometrical manner.

If random values are assigned to u, v and w, the law of aberration resulting from these equations will be a complicated one; but if u, v and w are such that udx + vdy + wdz is an exact differential, we have,

whence, denoting by the suffixes 1, 2 the values of the variables belonging to the first and second limits respectively, we obtain

If the motion of the ether be such that udx + vdy + wdz is an exact differential for one system of rectangular axes, it is easy to prove, by the transformation of co-ordinates, that it is an exact differential for any other system. Hence the formula (6) will hold good, not merely for light propagated in the direction first considered, but for light propagated in any direction, the direction of propagation being taken in each case for the axis of z. If we assume that udx+vdy+wdz is an exact differential for that part of the motion of the ether which is due to the motion of translation of the earth and planets, it does not therefore follow that the same is true for that part which depends on their motions of rotation. Moreover, the diurnal aberration is too small to be detected by observation, or at least to be measured with any accuracy, and I shall therefore neglect it.

It is not difficult to shew that the formulæ (6) lead to the known law of aberration. In applying them to the case of a star, if we begin the integrations in equations (5) at a point situated at such a distance from the earth that the motion of the ether, and consequently the resulting change in the direction of the light, is insensible, we shall have u1 =0, v1=0; and if, moreover, we take the plane az to pass through the direction of the earth's motion, we shall have

1

that is, the star will appear displaced towards the direction in which the earth is moving, through an angle equal to the ratio of the velocity of the earth to that of light, multiplied by the sine of the angle between the direction of the earth's motion and the line joining the earth and the star.

ADDITIONAL NOTE.

[In what precedes waves of light are alone considered, and the course of a ray is not investigated, the investigation not being required. There follows in the original paper an investigation having for object to shew that in the case of a body like the moon or a planet which is itself in motion, the effect of the distortion of the waves in the neighbourhood of the body in altering the apparent place of the body as determined by observation is insensible. For this, the orthogonal trajectory of the wave in its successive positions from the body to the observer is considered, a trajectory which in its main part will be a straight line, from which it will not differ except in the immediate neighbourhood of the body and of the earth, where the ether is distorted by their respective motions. The perpendicular distance of the further extremity of the trajectory from the prolongation of the straight line which it forms in the intervening quiescent ether is shewn to subtend at the earth an angle which, though not actually 0, is so small that it may be disregarded.

The orthogonal trajectory of a wave in its successive positions does not however represent the course of a ray, as it would do if the ether were at rest. Some remarks made by Professor Challis in the course of discussion suggested to me the examination of the path of a ray, which in the case in which uda+vdy+wdz is an exact differential proved to be a straight line, a result which I had not foreseen when I wrote the above paper, which I may mention was read before the Cambridge Philosophical Society on the 18th of May, 1845 (see Philosophical Magazine, vol. XXIX., p. 62). The rectilinearity of the path of a ray in this case, though not expressly mentioned by Professor Challis, is virtually contained in what he wrote. The problem is rather simplified by introducing the consideration of rays, and may be treated from the beginning in the following manner.

The notation in other respects being as before, let a', B' be the small angles by which the direction of the wave-normal at the point (x, y, z) deviates from that of Oz towards Ox, Oy, respectively, so that a, B' are the complements of a, B, and let a,, ß, be the inclinations to Oz of the course of a ray at the same point. By compounding the velocity of propagation through the ether with the velocity of the ether we easily see that

Let us now trace the changes of a, B, during the time dt. These depend first on the changes of a', B', and secondly on those of u, v.

As regards the change in the direction of the wave-normal, we notice that the seat of a small element of the wave in its successive positions is in a succession of planes of particles nearly parallel to the plane of x, y. Consequently the direction of the element of the wave will be altered during the time dt by the motion of the ether as much as a plane of particles of the ether parallel to the plane of the wave, or, which is the same to the order of small quantities retained, parallel to the plane xy. Now if we consider a particle of ether at the time t having for coordinates x, y, z, another at a distance de parallel to the axis of x, and a third at a distance dy parallel to the axis of y, we see that the displacements of these three particles parallel to the axis of z during the time dt will be

and dividing the relative displacements by the relation distances, we have dw/dx.dt, dw/dy. dt for the small angles by which the normal is displaced, in the planes of xz, yz, from the axes x, y, so that

We have seen already that the changes of u, v are du/dz. Vdt, dv/dz. Vdt, so that

Hence, provided the motion of the ether be such that

udx+vdy+wdz

is an exact differential, the change of direction of a ray as it travels along is nil, and therefore the course of a ray is a straight line notwithstanding the motion of the ether. The rectilinearity of propagation of a ray of light, which à priori would seem very likely to be interfered with by the motion of the ether produced by the earth or heavenly body moving through it, is the tacit assumption made in the explanation of aberration given in treatises of Astronomy, and provided that be accounted for the rest follows as usual. It follows further that the angle subtended at the earth by the perpendicular distance of the point where a ray leaves a heavenly body from the straight line prolonged which represents its course through the intervening quiescent ether, is not merely too small to be observed, but actually nil.]

* To make this explanation quite complete, we should properly, as Professor Challis remarks, consider the light coming from the wires of the observing telescope, in company with the light from the heavenly body.

[From the Philosophical Magazine, Vol. XXVIII. p. 76. (Feb. 1846.)]

ON FRESNEL'S THEORY OF THE ABERRATION OF LIGHT.

2

THE theory of the aberration of light, and of the absence of any influence of the motion of the earth on the laws of refraction, &c., given by Fresnel in the ninth volume of the Annales de Chimie, p. 57, is really very remarkable. If we suppose the diminished velocity of propagation of light within refracting media to arise solely from the greater density of the ether within them, the elastic force being the same as without, the density which it is necessary to suppose the ether within a medium of refractive index μ to have is u2, the density in vacuum being taken for unity. Fresnel supposes that the earth passes through the ether without disturbing it, the ether penetrating the earth quite freely. He supposes that a refracting medium moving with the earth carries with it a quantity of ether, of density μ2-1, which constitutes the excess of density of the ether within it over the density of the ether in vacuum. He supposes that light is propagated through this ether, of which part is moving with the earth, and part is at rest in space, as it would be if the whole were moving with the velocity of the centre of gravity of any portion of it, that is, with a velocity (1-2) v, v being the velocity of the earth. It may be observed however that the result would be the same if we supposed the whole of the ether within the earth to move together, the ether entering the earth in front, and being immediately condensed, and issuing from it behind, where it is immediately rarefied, undergoing likewise sudden condensation or rarefaction in passing from one refracting medium to another. On this supposition, the evident condition that a mass v of the ether must pass in a unit of time across a plane of area unit