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Mr. G. G. Stokes on the Aberration of Light.

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 which have reached the neighbourhood of the observer, the æther near him being supposed to be at rest relatively to him. If the æther 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 æther 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 æther. Neither of these suppositions is however true, for the æther 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 æther whose co-ordinates are $x,y,z$, and let $V$ be the velocity of light supposing the æther at rest. In consequence of the distance of the heavenly bodies, it will be quite unnecessary to consider any waves but those which are plane, except in so far as they are distorted by the motion of the æther. Let the axis of $z$ be taken in, or nearly in the direction of propagation of the wave considered, so that the equation to the wave's front at any time will be

$z=C+VT+\zeta$

(1.)

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

Denoting by $\alpha ,\beta ,\gamma$ 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,

$\cos \alpha =-{\frac {d\zeta }{dx}},\ \cos \beta =-{\frac {d\zeta }{dy}},\ \cos \gamma =1;$

(2.)

and if we take a length $Vdt$ along this normal, the co-ordinates of its extremity will be

$x-{\frac {d\zeta }{dx}}Vdt,\ y-{\frac {d\zeta }{dy}}Vdt,\ z+Vdt$

If the æther 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 $t+dt$,