Page:StokesAberration1846.djvu/3

Theory of the Aberration of Light. uninfluenced by the motion of the earth. The method which I employ will, I hope, be found simpler than Fresnel's; besides it applies easily to the most general case. Fresnel has not given the calculation for reflexion, but has merely stated the result; and with respect to refraction, he has only considered the case in which the course of the light within the refracting medium is in the direction of the earth's motion. This might still leave some doubt on the mind, as to whether the result would be the same in the most general case.

If the æther were at rest, the direction of light would be that of a normal to the surfaces of the waves. When the motion of the æther is considered, it is most convenient to define the direction of light to be that of the line along which the same portion of a wave moves relatively to the earth. For this is in all cases the direction which is ultimately observed with a telescope furnished with cross wires. Hence, if $$A$$ is any point in a wave of light, and if we draw $$A B$$ normal to the wave, and proportional to $$V$$ or $$\tfrac{V}{\mu}$$, according as the light is passing through vacuum or through a refracting medium, and if we draw $$B C$$ in the direction of the motion of the æther, and proportional to $$v$$ or $$\tfrac{v}{\mu^{2}}$$, and join $$A C$$, this line will give the direction of the ray. Of course, we might equally have drawn $$A D$$ equal and parallel to $$B C$$ and in the opposite direction, when $$D B$$ would have given the direction of the ray.

Let a plane $$P$$ be drawn perpendicular to the reflecting or refracting surface and to the waves of incident light, which in this investigation may be supposed plane. Let the velocity $$v$$ of the æther in vacuum be resolved into $$p$$ perpendicular to the plane $$P$$, and $$q$$ in that plane; then the resolved parts of the velocity $$\tfrac{v}{\mu^{2}}$$ of the æther within a refracting medium will be $$\frac{p}{\mu^{2}},\frac{q}{\mu^{2}}$$. Let us first consider the effect of the velocity $$p$$.

It is easy to see that, as far as regards this resolved part of the velocity of the æther, the directions of the refracted and reflected waves will be the same as if the æther were at rest. Let $$BAC$$ (fig. I) be the intersection of the refracting surface and the plane $$P$$; $$D A E$$ a normal to the refracting surface; $$AF, AG, AH$$ normals to the incident, reflected and refracted waves. Hence $$A F, A G, A H$$ will be in the plane $$P$$, and

$\angle GAD=FAD,\ \mu\sin HAE=\sin FAD$

Take

$AG=AF,\ AH=\frac{1}{\mu}AF$