Page:SearleEllipsoid.djvu/5

Thus, as Prof. Morton has also shown by the same method,

$\mathbf {\Psi } =\int _{\lambda }^{\infty }{\frac {q\alpha d\lambda }{2\mathrm {K} {\sqrt {\left(a^{2}+\alpha \lambda \right)\left(b^{2}+\lambda \right)\left(c^{2}+\lambda \right)}}}}$.

(12)

Now I have shown {§21} that if there is a surface A carrying a charge $q$, and any surface B is found for which $\mathbf {\Psi }$ is constant, then a charge $q$ placed upon B and allowed to acquire an equilibrium distribution will produce at all points not inside B the same effect as the charged surface A.

Hence the ellipsoid (11) when carrying a charge $q$ produces at all points not inside itself exactly the same disturbance as the ellipsoid $a,b,c$ with the same charge.

If we make $a=b=c=0$, the surfaces of equal "convection potential" are the ellipsoids given by

${\frac {x^{2}}{\alpha }}+y^{2}+z^{2}{=}\lambda$.

They are therefore all similar to each other. Thus the ellipsoid of this form produces exactly the same effect as a point-charge at its centre, and thus an ellipsoid of this form takes the place of the sphere in electrostatics. An ellipsoid with its axes in the ratios ${\sqrt {\alpha }}:1:1$ I have called a Heaviside Ellipsoid, since Mr. Heaviside^[1] was the first to draw attention to its importance in the theory of moving charges. Whatever be the ratios $a:b:c$, the equipotential surfaces

1. 'Electrical Papers,' vol. ii. p. 514.