# 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.