Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/461

 easily be derived by considering that the equation of the rectangular hyperbola

(in the plane of the variables ) is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters. The equations of transformation are thus found to be

where denotes a constant. The simpler equations previously given by Lorentz may evidently be derived from these by writing for, and neglecting powers of  above the first. By an obvious extension of the equations given by Lorentz for the electric and magnetic forces, it is seen that the corresponding equations in the present transformation are

The connexion between and  may be obtained in the following way. It is assumed that if a charge is attached to a particle which occupies the position  at the instant, an equal charge will be attached to the corresponding point  at the corresponding instant , in the transformed system; so that a charge  attached to an adjacent particle  at the instant  will give rise in the derived system to a charge  at the place at the instant