Page:CunninghamPrinciple.djvu/3

540 According to this transformation, as a more geometrical correspondence, the length of a line in the direction of the axis of x moving with the axes A', as measured in the coordinates x', y', z', t', is greater than its length measured in the coordinates x, y, z, t in the ratio $$1:\left(1-v^{2}/c^{2}\right){}^{\frac{1}{2}}$$, so that Lorentz's hypothesis of the reduction in the dimensions of a body when it moves relatively to an observer is reduced by this geometrical correspondence to the assumption that in the variables associated with axes moving with it its shape is unaltered — an assumption suggested by the fact that the electromagnetic equations referred to those variables are independent of the motion through the aether, and by the attempt to form a purely electromagnetic theory of matter. Thus if the single electron at rest has a spherical configuration, and there are no other than electromagnetic forces, we should expect it in motion to have a spherical configuration when measured by the variables x' y' z' which means that as measured by the variables x, y, z it will have the spheroidal shape as suggested by Lorentz.

The electron as conceived by Abraham, on the other hand, is spherical always as regards the variables x, y, z, and a prolate spheroid as regards x' y' z' the ratio of the axes being $$1:\left(1-v^{2}/c^{2}\right){}^{\frac{1}{2}}$$.

In either case the electromagnetic mass is defined as the ratio of the external mechanical force on the electron to the acceleration of the centre of the electron, and Abraham develops two expressions for the longitudinal mass, i. e. for the ratio of the force in the direction of motion to the acceleration in the same direction, viz.: $$\tfrac{dG}{dv}$$ and $$\tfrac{1}{v}\frac{dW}{dv}$$ for the case of the so-called quasistationary motion, G being the electromagnetic momentum and W the electromagnetic energy. For the latter case these two expressions are proved identical, but for the Lorentz electron they are not equal, and Abraham deduces that W cannot be the whole energy of the electron. But the fact is that in this case the mass as above defined is not equal to $$\tfrac{1}{v}\frac{dW}{dv}$$, this expression being obtained on the assumption that the electron is "rigid" (Theorie der Elek. ii. p. 155). If the change in the shape of the electron with the change in velocity is taken into account, it will be found that the mass as obtained from the change in momentum is identical with the mass as obtained from the change in the energy, as it clearly must be, since a quantity defined in a perfectly definite manner cannot from consistent equations be shown to have two different values.