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corrected for his observed relative velocity, give the same configuration as the observations of S uncorrected, and the latter is a permanent standard so long as his velocity is uniform. But for E the whole universe is rotated in space through an angle θ + α - χ. This rotation is precisely that which is required to enable the two observers to identify the same star as occupying the position which each observer imagines to represent the apex of the motion of E relative to S, and thus it could not be detected even though the two observers were in communication. The difficulty discussed in § 11 arises from the fact that the observed velocity of E relative to S and the observed velocity of S relative to E are not in the same straight line. We find that

$\tan(\chi -\theta )={\frac {\sin \alpha \left(\cos \beta _{0}+\cos \beta _{1}\right)}{\cos \alpha \left(1+\cos \beta _{0}\cos \beta _{1}\right)-\sin \beta _{0}\sin \beta _{1}}},$

which can be reduced to the form

$\tan {\frac {1}{2}}(\chi -\theta )={\frac {\cos {\frac {1}{2}}\left(\beta _{1}-\beta _{0}\right)}{\cos {\frac {1}{2}}\left(\beta _{1}+\beta _{0}\right)}}\tan {\frac {1}{2}}\alpha$

Thus χ - θ is not equal to α, as it would be in ordinary kinematics.

14. We have thus given a perfectly precise meaning to the application of the principle of relativity to stellar aberration. In doing so we have excluded everything which does not belong to the purely optical aspect of the problem. But stellar aberration is not a purely optical problem. For in practice we do not actually observe the apparent motion of the Sun and use the result to correct our observed positions of the stars. The motion which we do use is derived by calculation from the theory of gravitation. Hence, if we are to be consistent, we must regard Keplerian motion as an appearance, not as a reality. And here we come in contact with the general problem of the dynamics of the electron, which in the historical sense is responsible for the introduction of the principle of relativity. In this wider problem there is a profound modification of the notion of mass, in addition to the transformation which we have admitted in the time and the spatial coordinates. The result of the work of Lorentz and others is to show that these transformations suffice to explain the complete compensation of effects arising from the motion of any system through space over the whole field of electrodynamics as well as of optics. The same will be true of gravitation if gravity can be expressed in terms of electrodynamic entities. If, on the other hand, the nature of gravity were essentially not electrodynamic, it would be possible for some deviation from the accepted laws of dynamics, owing to a motion of translation through space, to be revealed by direct astronomical observations. The possibility of such deviations and of the compensations, partial or complete, which may accompany them, places some astronomical problems in a new position. Thus Poincaré has recently reconsidered^[1] the

1. C.R., cxl. p. 1504.