Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/441

 kinetic potential which concerns any one of them—say, e—may be written

where and  denote potential functions, defined by the

denoting the volume-density of electric charge, and its velocity, and the integration being taken over all space.

We shall now reject Clausius' assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other only through the mediation of an aether which fills all space, and satisfies Maxwell's equations, This modification may be effected in Clausius' theory without difficulty; for, as we have seen, if the state of Maxwell's aether at any point is defined by the electric vector and magnetic vector, these vectors may be expressed in terms of potentials  and  by the equations

and the functions and  may in turn be expressed in terms of the electric charges by the equations

where the bars indicate that the values of and  refer to the instant. Comparing these formulae with those given above for Clausius' potentials, we see that the only change which it is necessary to make in Clausius' theory is that of retarding the potentials in the way indicated by L. Lorenz. The electric and magnetic forces, thus defined in terms of the