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

æther may be neglected, then to a point near the earth where we may still neglect the motion of the æther, and lastly to the point of the earth's surface at which the planet is viewed. For the first part we shall have $u_{2}=0,\ v_{2}=0$, and $u_{1},v_{1}$ will be the resolved parts of the planet's velocity. The increments of $\alpha$ and $\beta$ for the first interval will be, therefore, $-{\tfrac {u_{1}}{V}},\ {\tfrac {v_{1}}{V}}$. For the second interval $\alpha$ and $\beta$ will remain constant, while for the third their increments will be ${\tfrac {u_{2}}{V}},\ {\tfrac {v_{2}}{V}}$, just as in the case of a star, $u_{2}$ and $v_{2}$ being now the resolved parts of the earth's velocity.

Fig. 1^[1]

Fig. 2

Fig. 1 represents what is conceived to take place. $P$ is the planet in the position it had when the light quitted it; $E$ the earth in the position it has when the light reaches it. The lines $ab,cd$, &c. represent a small portion of a wave of light in its successive positions. The arrows represent the directions in which $P$ and $E$ may be conceived to move. The breadth $ab$ is supposed to be comparable to the breadth of a telescope. In fig. 2, $pmne$ represents an orthogonal trajectory to the surfaces $ab,cd$, &c. ; $p$ is the point of the planet from which the light starts, $e$ the point of the earth which it reaches. The trajectory $pmne$ may be considered a straight line, except near the ends $p$ and $e$, where it will be a little curved, as from $p$ to $m$ and from $e$ to $n$. The curvature at $e$ will have the same effect on the apparent position of the planet as it would have on that of a star in the same direction : as to the curvature at $p$, if we draw $pq$ perpendicular to $mn$ produced, the curvature will have the effect of causing $p$ to be seen as if it were at $q$. Now the angle between the tangents at $p$ and $m$ being that through which a star in the direction of $e$ is displaced by aberration to an observer at $p$, and the distance

1. The lines towards $P$ in fig. 1. should lean in the opposite direction.