Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/362

 Now, let denote coordinates relative to axes which are parallel to the axes, and which move with the charged bodies; then , is a function of  only; so we have

and the preceding equation is readily seen to be equivalent to

where denotes. But this is simply Poisson's equation, with substituted for ; so the solution may be transcribed from the known solution of Poisson's equation: it is

the integrations being taken over all the space in which there are moving charges; or

If the moving system consists of a single charge at the point , this gives

where.

It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being

The electric force is radial, its magnitude being

The fact that the electric vector due to a moving point-charge is everywhere radial led Heaviside to conclude that the same solution is applicable when the charge is distributed over