Page:SearleEllipsoid.djvu/5

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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.

Searle1.png

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.