Thus in both cases the corrected expression for the longitudinal mass as derived from the energy gives the same result as that obtained from the momentum, and no other forces other than electromagnetic come into play.
On the other hand, since from what has been said above it appears that the electron will naturally retain the spherical shape as measured by the variables associated with the moving axes, it appears that some extraneous forces would be required to cause it to retain the spherical shape to an observer remaining at rest.
It is perhaps worth noting that "the principle of relativity" propounded by Bucherer in the Phil. Mag. of April 1907 is in essence identical with the statement made in the beginning of this paper. The principle referred to may be stated thus : that in the sequence of electromagnetic phenomena the giving of an additional uniform translational velocity v to the whole system of electric and magnetic bodies will not affect the phenomena observed if this velocity v is at the same time given to the observer. The transformation of space and time variables mentioned above shows a means of explaining this dependence of the electromagnetic phenomena on relative motion only; and conversely it is a comparatively simple matter to show that it is the only means. For it is required, among other things, to explain how a light-wave travelling outwards in all directions with velocity C relative to an observer A, may at the same time be travelling outwards in all directions with the same velocity relative to an observer B moving relative to A with velocity v. This can clearly not be done without some transformation of the space and time variables of the two observers.
Suppose two observers A, B, to be situated momentarily in the same spot, and let B be moving relatively to A with velocity v (measured by A in his own system of space and time). Let the direction of motion of B be A's axis of x, and let the instant of coincidence be A's time t=0.
Suppose axes of ξ η ζ to be B's system of coordinates moving with him with velocity v relative to A's axes of x y z, and coinciding with them at t=0.
Associated with a given point at a given time as marked by the values (x, y, z, t) will be unique values of (ξ, η, ζ, τ), τ being B's measure of the interval elapsed from the time of his coincidence with A, and conversely. There must therefore be a linear transformation from the variables (x, y, z, t) to (ξ, η, ζ, τ).
Consider now points on the axis of x (or ξ). Then the
