Page:MortonCharge.djvu/3

182 with uniform velocity u parallel to the axis of z, and that the field has assumed its steady configuration. We shall denote $$1-\frac{u^{2}}{V^{2}}$$ by k², V being the velocity of light. Then since we have a steady state,

$$\frac{d}{dt}=-u\frac{d}{dz}$$.

Also, since each element of charge produces a magnetic field with no z-component, we have $$\gamma=0$$ in the general case also. Using these two data, the equations connecting the displacement (f, g, h) and the magnetic force (α, β, γ) become

These equations together with

$$\frac{df}{dx}+\frac{dg}{dy}+\frac{dh}{dz}=0$$

are satisfied by

where φ is any function satisfying

$$\frac{d^{2}\phi}{dx^{2}}+\frac{d^{2}\phi}{dy^{2}}+k^{2}\frac{d^{2}\phi}{dz^{2}}=0$$.