Page:PlummerAberration.djvu/9

260 V (supposed constant) of the system. If V' is the resultant and U the velocity of light, we have

$ES=U(t-T),\ SS'=V'(t-T),$

E (fig. 3) being the position of the Earth at time t, S the position of the star at time T, ES' the observed direction, and SS' the virtual displacement of the star. But this displacement is compounded of two, SS0 and S0S', such that

$SS_{0}=V(t-T),\ S_{0}S'=v(t-T).$

Let ES0 = Uτ. If then we ignore the velocity V, as in practice we do, and assume a virtual star in the permanently displaced position SO, we shall infer from observation a fictitious velocity of the Earth v' such that

$S_{0}S'=v(t-T)=v'\tau$

or

$v'/v=(t-T)/\tau=ES/ES_{0}=U/\sqrt{U^{2}+V^{2}+2UV\ \cos\phi},$

where φ is the angle between the true direction of the star and the direction of the secular motion of the solar system. Otherwise expressed, the result is as if the constant of aberration for the given star is changed in the ratio of $$U:\sqrt{U^{2}+V^{2}+2UV\ \cos\phi}$$. This is an elementary deduction of Villarceau’s result, and on the assumed premises no other effect is to be expected.

The subject of secular aberration was discussed by Seeliger, who arrived at results which are quite inadmissible. His expressions for the effects of aberration in R.A. and declination become infinite at the pole, whereas it is obvious that there is no singularity in the neighbourhood of this point in the sky. There is no special connection between the pole and the direction of the operative vectors considered above, and such results can only be due to faulty analysis. It would have been unnecessary to refer