[From the Philosophical Magazine, Vol. xxvII. p. 9. (July, 1845.)]

ON THE ABERRATION OF LIGHT.

THE general explanation of the phenomenon of aberration is so simple, and the coincidence of the value of the velocity of light thence deduced with that derived from the observations of the eclipses of Jupiter's satellites so remarkable, as to leave no doubt on the mind as to the truth of the explanation. But when we examine the cause of the phenomenon more closely, it is far from being so simple as it appears at first sight. On the theory of emissions, indeed, there is little difficulty; and it would seem that the more particular explanation of the cause of aberration usually given, which depends on the consideration of the motion of a telescope as light passes from its object-glass to its cross wires, has reference especially to this theory; for it does not apply to the theory of undulations, unless we make the rather startling hypothesis that the luminiferous ether passes freely through the sides of the telescope and through the earth itself. The undulatory theory of light, however, explains so simply and so beautifully the most complicated phenomena, that we are naturally led to regard aberration as a phenomenon unexplained by it, but not incompatible with it.

The object of the present communication is to attempt an explanation of the cause of aberration which shall be in accordance with the theory of undulations. I shall suppose that the earth and the planets carry a portion of the ether along with them so that the ether close to their surfaces is at rest relatively to those surfaces, while its velocity alters as we recede from the surface, till, at no great distance, it is at rest in space. According to the undulatory theory, the direction in which a heavenly body is seen

is normal to the fronts of the waves which have emanated from it, and have reached the neighbourhood of the observer, the ether near him being supposed to be at rest relatively to him. If the ether in space were at rest, the front of a wave of light at any instant being given, its front at any future time could be found by the method explained in Airy's tracts. If the ether were in motion, and the velocity of propagation of light were infinitely small, the wave's front would be displaced as a surface of particles of the ether. Neither of these suppositions is however true, for the ether moves while light is propagated through it. In the following investigation I suppose that the displacements of a wave's front in an elementary portion of time due to the two causes just considered take place independently.

Let u, v, w be the resolved parts along the rectangular axes of x, y, z, of the velocity of the particle of ether whose co-ordinates are x, y, z, and let V be the velocity of light supposing the ether at rest. In consequence of the distance of the heavenly bodies, it will be quite unnecessary to consider any waves except those which are plane, except in so far as they are distorted by the motion of the ether. Let the axis of z be taken in, or nearly in the direction of propagation of the wave considered, so that the equation of a wave's front at any time will be

C being a constant, t the time, and a small quantity, a function and t. Since u, v, w and are of the order of the aberration, their squares and products may be neglected.

of x, y

Denoting by a, B, y the angles which the normal to the wave's front at the point (x, y, z) makes with the axes, we have, to the first order of approximation,

and if we take a length Vdt along this normal, the co-ordinates

of its extremity will be

d dx

dr
Vdt, z+ Vdt.

Vdt, y dy

If the ether were at rest, the locus of these extremities would be the wave's front at the time t+dt, but since it is in motion, the

co-ordinates of those extremities must be further increased by udt, vdt, wdt. Denoting then by x', y', z' the co-ordinates of the point of the wave's front at the time t + dt which corresponds to the point (x, y, z) at the time t, we have

and eliminating x, y and z from these equations and (1), and denoting by f(x, y, t), we have for the equation to the wave's front at the time t + dt,

or, expanding, neglecting dt2 and the square of the aberration, and suppressing the accents of x, y and z,

.....

z = C + Vt + G + (w + V) dt.......... .(3). But from the definition of it follows that the equation to the wave's front at the time t+dt will be got from (1) by putting t+dt for t, and we have therefore for this equation

Comparing the identical equations (3) and (4), we have

.(4).

This equation gives 5=fwdt; but in the small term & we may replace fwdt by fwdz + V: this comes to taking the approximate

value of z given by the equation z = C+ Vt instead of t for the parameter of the system of surfaces formed by the wave's front in its successive positions. Hence equation (1) becomes

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 formulæ (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 xz to pass through the direction of the earth's motion, we shall have

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 udx + 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.