Page:Deux Mémoires de Henri Poincaré.djvu/6

 simplification. Not having noticed it, I did not succeed in obtaining the exact invariance of the equations; my formulas remained encumbered with certain terms which should have disappeared. These terms were too small to have an appreciable effect on the phenomena and I could thus explain the independence of the earth's motion that was revealed by observations, but I did not establish the principle of relativity as rigorously and universally true.

Poincaré, on the contrary, obtained a perfect invariance of the equations of electrodynamics, and he formulated the "postulate of relativity", terms which he was the first to employ. Indeed, stating from the point of view that I had missed, he found the formulas (4) and (7). Let us add that by correcting the imperfections of my work he never reproached me for them.

I can not explain here all the beautiful results obtained by Poincaré. Let us insist however on some points. Initially, he was not satisfied to show that the transformations of relativity leave intact the form of the electromagnetic equations. He explains the success of substitutions by noticing that these equations can be put in the form of the principle of least action and that the fundamental equation which expresses this principle, as well as the operations by which we deduce the field equations, are the same in systems x, y, z, t and $$x', y', z', t'$$.

In the second place, in accordance with the title of his paper, Poincaré particularly considers the way in which the deformation of a moving electron occurs, comparable with that of the arms of the device of Mr. Michelson, which is required by the postulate of relativity. Two different assumptions had been proposed on this subject. According to both an electron, presumably spherical in the state of rest, would change by a translation into an oblated ellipsoid of revolution, the axis of symmetry coincide with the direction of motion and the ratio of this axis to the diameter of the equator being given by $$\sqrt{1-v^{2}},$$, if v is the velocity. But the assumptions differed between them with regard to the length of the axes and consequently the volume of the electron. While I had been led to admit that the radius of the equator remains equal to that of the original sphere, Mr. Bucherer and Mr. Langevin rather wanted to assign a constant size with volume. The first assumption corresponds to l = 1, the second with $$kl^3 =1$$. Let us immediately add that the first value is the only one which is compatible with the postulate of relativity.