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388 On the bearing of the Principle of Relativity on Gravitational Astronomy. By W. de Sitter, Assoc. R.A.S.

1. The principle of relativity, first developed in connection with the electromagnetic theory of light, has in recent years been more and more considered as of universal application, and the claim has been made that the whole of our physical sciences should be framed in conformity with it. In this paper are considered some of the results of introducing the principle in the laws of planetary motion. Poincaré has made the remark that some of the difficulties of the problem of three bodies are due to the fact that in the undisturbed motion the nodes and perihelia are fixed. Now, with a law of attraction which is in conformity with the principle of relativity, the perihelion has a motion even in the problem of two bodies, and it is probable that such questions as the existence of periodic solutions and the convergence of series entirely change their aspect. In the present paper, however, these interesting mathematical questions are not touched upon, and I consider the problem exclusively from the point of view of the practical astronomer, investigating only such effects as may be expected to yield the possibility of an empirical verification of the principle.

All natural phenomena are described in terms of four variable quantities; three "coordinates" $$x, y, z,$$ and the "time" $$t$$. If we introduce other variables $$x', y', z', t',$$ the formulæ describing any event will generally be altered. Also the physical laws, which are differential equations defining $$x, y, z,$$ as functions of $$t$$ and certain constants, are in general altered by a change of the system of reference. Physical science has, however, been so built up that there are certain transformations which leave the form of the equations unaltered. In the classical, or Newtonian, mechanics these are all orthogonal transformations of the three coordinates $$x, y, z,$$ combined with an arbitrary uniform velocity of these axes and an arbitrary change of the units of length and time.

The principle of relativity can be enunciated as the postulate