Page:StokesAberration1845.djvu/4

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

one; but if $u,v$ and $w$ are such that $udv+vdy+wdz$ is an exact differential, we have

${\frac {dw}{dx}}={\frac {du}{dz}},\ {\frac {dw}{dy}}={\frac {dv}{dz}}$

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

$\alpha _{2}-\alpha _{1}={\frac {u_{2}-u_{1}}{V}},\ \beta _{2}-\beta _{1}={\frac {v_{2}-v_{1}}{V}}$

(6.)

If the motion of the æther be such $udx+vdy+wdz$ 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 formulae (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 æther which is due to the motions 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 show that the formulae (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 æther, and consequently the resulting change in the direction of the light, is insensible, we shall have $u_{1}=0,\ v_{1}=0$; and if, moreover, we take the plane $xz$ to pass through the direction of the earth's motion, we shall have

$v_{2}=0,\ \beta _{2}-\beta _{1}=0,$

and

$\alpha _{2}-\alpha _{1}={\frac {u_{2}}{V}}$

that is, the star will appear to be 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.

In considering the effect of aberration on a planet, it will be convenient to divide the integrations in equation (5.) into three parts, first integrating from the point considered on the surface of the planet to a distance at which the motion of the