# Page:SearleEllipsoid.djvu/5

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

 ${\displaystyle \mathbf {\Psi } =\int _{\lambda }^{\infty }{\frac {q\alpha d\lambda }{2\mathrm {K} {\sqrt {\left(a^{2}+\alpha \lambda \right)\left(b^{2}+\lambda \right)\left(c^{2}+\lambda \right)}}}}}$. (12)

Now I have shown {§21} that if there is a surface A carrying a charge ${\displaystyle q}$, and any surface B is found for which ${\displaystyle \mathbf {\Psi } }$ is constant, then a charge ${\displaystyle q}$ 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 ${\displaystyle q}$ produces at all points not inside itself exactly the same disturbance as the ellipsoid ${\displaystyle a,b,c}$ with the same charge.

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

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

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 ${\displaystyle {\sqrt {\alpha }}:1:1}$ 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 ${\displaystyle a:b:c}$, the equipotential surfaces

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