Page:PlummerAberration.djvu/11

suppose that he is moving relatively to S in the direction of some point E' on the great circle SE; and if he is to deduce the position P₀ in space from the observed position of the star P₁, the point E' must lie also on the great circle P₀P₁. Since P₀, P₁, E' are points on the sides of the triangle SPE, and S, E, E' are points on the sides of the triangle PP₀P₁, we have

${\begin{array}{ccccc}{\frac {\sin SE'}{\sin EE'}}&\cdot &{\frac {\sin EP_{1}}{\sin P_{1}P}}&\cdot &{\frac {\sin PP_{0}}{\sin P_{0}S}}=1\\\\{\frac {\sin P_{0}E'}{\sin P_{1}E'}}&\cdot &{\frac {\sin P_{1}E}{\sin PE}}&\cdot &{\frac {\sin PS}{\sin P_{0}S}}=1.\end{array}}$

Let

$PS=\phi _{0},\ P_{0}S=\phi '_{0},\ PP_{0}=\phi _{0}\ \phi '_{0}$

$PE=\phi _{1},\ P_{1}E=\phi '_{1},\ PP_{1}=\phi _{1}-\phi '_{1}.$

Hence

${\frac {\sin \ SE'}{\sin \ EE'}}={\frac {\sin \left(\phi _{1}-\phi '_{1}\right)}{\sin \phi '_{1}}}\cdot {\frac {\sin \phi '_{0}}{\sin \left(\phi _{0}-\phi '_{0}\right)}}$

${\frac {\sin P_{0}E'}{\sin P_{1}E'}}={\frac {\sin \phi _{1}}{\sin \phi '_{1}}}\cdot {\frac {\sin \phi '_{0}}{\sin \phi {}_{0}}}.$

The first of these shows that

${\frac {\sin \left(\phi _{1}-\phi '_{1}\right)}{\sin \phi '_{1}}}\div {\frac {\sin \left(\phi _{0}-\phi '_{0}\right)}{\sin \phi '_{0}}}=const.$

for all stars, which happens to be true according to the ordinary theory of aberration. But it is not true according to the "relativity" law of aberration, which gives

${\frac {\sin \left(\phi -\phi '\right)}{\sin \phi '}}={\frac {\sin \beta +(1-\cos \beta )\cos \phi }{\cos \beta }}.$

In fact it appears that a law of aberration, which would explain the absence of a visible effect arising from the secular motion by supposing that the corrected observation coincides in space with the standard observation, would require simultaneously

$\sin(\phi -\phi ')/\sin \phi '=f_{1}(V),\ \sin(\phi -\phi ')/\sin \phi =f_{2}(V),$

and this is not possible. We have then to look in another direction for the explanation of the way in which the "relativity" law effects a compensation, and for the necessary hint I am indebted to Mr. Eddington, who very kindly supplied me with a particular case which was both simple and illuminating.

12. A general proof will now be given. Let the ether-velocity of the observer S be $V_{0}=U\ \sin \beta _{0}$, and of the observer E be $V_{1}=U\ \sin \beta _{1}$, and let the angle between them be α, being measured from V₀ towards V₁. The components of V₀ along and perpendicular