Page:Electromagnetic phenomena.djvu/9



units of time. In this last expression we may put for the differential coefficients their values at the point A.

In (17) we have now to replace $$\left[\varrho\right]$$ by

where $$\left[\frac{\partial\varrho}{\partial t}\right]$$ relates again to the time t0. Now, the value of t'  for which the calculations are to be performed having been chosen, this time t0 will be a function of the coordinates x, y, z of the exterior point P. The value of $$\left[\varrho\right]$$ will therefore depend on these coordinates in such a way that

by which (25) becomes

Again, if henceforth we understand by r'  what has above been called $$r_{0}^{'}$$, the factor $$\frac{1}{r'}$$ must be replaced by

so that after all, in the integral (17), the element dS is multiplied by

This is simpler than the primitive form, because neither r' , nor the time for which the quantities enclosed in brackets are to be taken, depend on x, y, z. Using (23) and remembering that $$\int\varrho\ dS=0$$, we get

a formula in which all the enclosed quantities are to be taken for the instant at which the local time of the centre of the particle is $$t'-\frac{r'}{c}$$.

We shall conclude these calculations by introducing a new vector $$\mathfrak{p}'$$, whose components are