Page:PlummerAberration.djvu/8

Jan. 1910. 9. The geometrical significance of the new law of aberration is interesting. In the first place we may consider the stereographic projection of the celestial sphere on the tangent plane perpendicular to the direction of motion. The form of the law of aberration

$\tan\frac{1}{2}\phi'=\sqrt{\frac{U-v}{U+v}}\tan\frac{1}{2}\phi$|undefined

shows that the effect of aberration is simply to alter the scale of the projection. But the stereographic projection is a conformal representation of the sphere. Hence actual configurations on the sphere are only changed conformally by the effect of aberration, or in other words any small area is altered only in size and not in shape. We know that aberration merely changes the scale of a photograph of a small part of the sky, and the truth of this fact now becomes independent of the velocity (however large) of the observer. Also stars which appear to lie on a circle at any one time will continue to do so permanently.

Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β. This means that we view the apparent celestial sphere from the centre. Then, to pass from the apparent direction of a star to its actual direction, we must imagine the sphere transformed into an ellipsoid by contracting its dimensions perpendicular to the axis along which motion takes place in the ratio of cos β : 1. The true direction will then be inferred by viewing the ellipsoid of revolution from the focus which is in advance of the centre. Conversely, we may interchange φ and φ' if we employ the other focus. We thus see that if we consider the true celestial sphere to undergo the contraction just specified, the apparent positions of the stars are given by viewing the ellipsoid from that focus which follows the centre.

10. Secular Aberration.— In what precedes, the question of all second-order effects connected with aberration has taken a new form. But we may return to the old order of ideas to consider briefly the subject of secular aberration which arises from the motion of the solar system through the ether of space, and which was first discussed by Villarceau. It is a very simple matter when regarded from the old point of view, but it has led to some apparent confusion. It has been seen (vol. lxix. p. 505) that if the light which leaves a star at the time T is observed at the time t, the observed direction of the star coincides with the actual direction of a fictitious body which is supposed to start from the position of the star at time T, and to continue in motion during the time (t — T) with a velocity equal and parallel to that of the Earth at the time t.

Now the velocity of the Earth is compounded of its velocity v relative to the centre of mass of the solar system and the velocity