Page:Deux Mémoires de Henri Poincaré.djvu/8

 and will inevitably produce the oblateness which is required by the principle of relativity.

After having found his supplementary force, Poincaré showed that the transformations of relativity do not change the form of the terms which it represents; thus he showed that arbitrary motions of a system of electrons can take place in the completely same manner in system x, y, z, t and in the system $$x', y', z', t'$$.

I already spoke about the necessity for posing l = 1 (constancy of the equatorial radius of the electron). I will not repeat here the demonstration given by Poincaré and I will only say that he showed the mathematical origin of this condition. One can consider all the transformations which are represented by formulas (1), with different values for speed -ε, and the corresponding values of k and l, this last coefficient has to be regarded as a function of ε; we can add to it other similar transformations which we deduce from (1) by changing the directions of the axes, and finally by arbitrary rotations. The postulate of relativity requires that all these transformations form a group and that is only possible if l has the constant value 1.

The "group of relativity" obtained, consists of linear substitutions which do not affect the quadratic form

$$x^2 + y^2 + z^2 - t^2$$

The paper ends with the application of the postulate of relativity on the phenomena of gravitation. Here, it is the question of finding the rule which determines the propagation of it, and the formulas which express the components of the force according to the coordinates and the speed, as well as of the attracted body as of the attracting body. By considering these questions, Poincaré starts by seeking the invariants of the group of relativity; indeed, it is clear that it must be possible to represent the phenomena by equations which contain only these invariants. However, the problem is undetermined. It is natural to admit that the propagation velocity is equal to that of light and that the variations of the law of Newton must be of second-order magnitude in respect to the velocities. But even with these restrictions, there is the choice between several assumptions, among which there are two that were especially indicated by Poincaré.

In this last part of the article one finds some new concepts which I must especially announce. Poincaré notices, for example, if