Page:StokesAberration1845.djvu/6

Mr. G. G. Stokes on the Aberration of Light. $$p m$$ being by hypothesis small (two or three radii of the planet suppose), it follows that the angle $$peq$$ is extremely small, and may be neglected. Hence a planet will appear to be displaced from the position which it had when the light left it, just as a star in the same direction is displaced. But besides this, the planet has moved from $$P$$ while the light has been travelling to $$E$$. These two considerations combined lead to the formula for aberration, which is applicable to the planets, as is shown in treatises on astronomy. The same reasoning which applies to a planet will apply equally to the sun, the moon, or a comet.

To give an idea of the sort of magnitudes neglected in neglecting $$pq$$, suppose $$pm$$ equal to the diameter of $$P$$, and suppose the curvature from $$p$$ to $$m$$ uniform. Let $$r$$ be the radius of $$P$$, $$v$$ its velocity, and $$R$$ the distance $$P E$$. The greatest possible value of the angle between the tangents at $$p$$ and $$m$$ is $$\tfrac{v}{V}$$. In this case we should have $$\angle pep=\tfrac{vr}{VR}=\tfrac{v}{V}D$$, $$D$$ being the semidiameter of $$P$$ as seen from $$E$$. Hence the angle $$peq$$ must be very much greater for the moon than for any other body of the solar system; for in the case of the planets the value of $$v$$ is in no instance double its value for the earth or moon, while their discs are very small compared with that of the moon; and in the case of the sun, although its disc is about as large as that of the moon, its velocity round the centre of gravity of the solar system is very small. It would indeed be more correct to suppose the sun's centre absolutely at rest, since all our measurements are referred to it, and not to the centre of gravity of the solar system. Taking then the case of the moon, and supposing $$\tfrac{v}{V}=\tfrac{20''}{180^{\circ}}\pi$$, $$D $$ =15′, we find that the angle $$peq$$ is about $$\tfrac{1}{11}$$th of a second, an insensible quantity.

If we suppose the whole solar system to be moving in space with a velocity comparable with that of the earth round the sun, it follows from the linearity of the equations employed, that we may consider this motion separately. It is easy to show, that as far as regards this motion, the sun, moon, and planets will come into the positions in which they are seen just at the instant that the light from them reaches the earth. With respect to the stars also, that part of the aberration which varies with the time of year, the only part which can be observed, will not be affected. If we suppose the æther which fills the portion of space occupied by the solar system to be moving in a current, with a velocity comparable with that of the earth in its orbit, the result will still be the same. For if