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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


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.