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224 a satisfactory explanation of the results of the Michelson-Morley experiment can be given on the supposition of a stationary aether. The results of the experiment seem at first sight to indicate that the aether is carried along with the earth, but on such a supposition a satisfactory explanation of aberration would still be wanting.

The theory of Lorentz's transformation has been developed very considerably by Einstein, Planck, and Minkowski. The transformation has become a powerful instrument of research, inasmuch as it provides a means of transition from the mathematical expressions of physical quantities connected with a system at rest to the corresponding quantities for a similar system in uniform motion. The transformation of Lorentz only enables us to pass from a system at rest to one in uniform motion, but it has been postulated that there is a more general transformation which can be applied to systems moving in a more general manner. The case of a system moving with a constant acceleration has, in fact, been discussed by Einstein.

The object of the present paper is to find all the transformations for which the electrodynamical equations are invariant. In the case of the simpler equations of the theory of electrons, it is proved that the transformations belong to a certain group which is isomorphic with the group of conformal transformations of a space of four dimensions. It is assumed, however, that the transformation is such that the total charge on a system of particles is unaltered.

I have great pleasure in thanking Mr. E. Cunningham for the stimulus which he gave to this research by the discovery of the formula of transformation in the case of an inversion in the four-dimensional space. These formulæ suggested the more general formulæ in terms of Jacobians, and the transition to the integral forms led at once to the present analysis. The two integral forms of the second order had been introduced previously by Mr. Hargreaves in another connection; they enable us to give a very concise expression of the electrodynamical equations which promises to be of considerable importance in future developments. The integrals that occur in the equations differ from the usual surface and volume integrals by the fact that the quantities in the integrand are calculated at different points of space at different times, these times being specified by an arbitrary known law.

The invariance of the two integral forms of the second order leads at once to the formulae of transformation of the electric and magnetic force,