# Page:SearleEllipsoid.djvu/10

The value of ${\displaystyle \mathbf {\Psi } }$ in terms of ${\displaystyle h}$ thus becomes

 ${\displaystyle \mathbf {\Psi } ={\frac {q\alpha }{\mathrm {K} }}\int _{h}^{\infty }{\frac {dh}{h^{2}-l^{2}}}}$. (17)

Equation (11) now becomes

 ${\displaystyle {\frac {x^{2}}{h^{2}}}+{\frac {\rho ^{2}\alpha }{h^{2}-l^{2}}}=1}$, (18)

so that instead of the cylindrical coordinates ${\displaystyle x}$ and ${\displaystyle \rho (={\sqrt {y^{2}+z^{2}}})}$ we, can take ${\displaystyle h}$ and ${\displaystyle \phi }$ where

 ${\displaystyle x=h\cos \phi ,\ \rho ={\frac {\sqrt {h^{2}-l^{2}}}{\sqrt {\alpha }}}\sin \phi }$. (19)

From (18) we have in terms of ${\displaystyle h}$ and ${\displaystyle \phi }$

${\displaystyle {\frac {dh}{dx}}={\frac {(h^{2}-l^{2})\cos \phi }{h^{2}-l^{2}\cos ^{2}\phi }},\ {\frac {dh}{d\rho }}={\frac {h{\sqrt {h^{2}-l^{2}}}\sin \phi {\sqrt {\alpha }}}{h^{2}-l^{2}\cos ^{2}\phi }}}$.

Hence

 ${\displaystyle \mathrm {E} _{1}=-{\frac {d\mathbf {\Psi } }{dh}}\cdot {\frac {dh}{dx}}={\frac {\alpha q\ \cos \phi }{K{\sqrt {h^{2}-l^{2}\cos ^{2}\phi )}}}}}$, (20)
 ${\displaystyle \mathrm {E} _{\rho }=-{\frac {1}{\alpha }}{\frac {d\mathbf {\Psi } }{dh}}\cdot {\frac {dh}{d\rho }}={\frac {qh\ \sin \phi {\sqrt {\alpha }}}{\mathrm {K} {\sqrt {h^{2}-l^{2}}}(h^{2}-l^{2}\cos ^{2}\phi )}}}$, (21)
 ${\displaystyle \mathrm {H} =\mathrm {K} u\mathrm {E} _{\rho }={\frac {quh\ \sin \phi {\sqrt {\alpha }}}{\sqrt {h^{2}-l^{2}(h^{2}-l^{2}\cos ^{2}\phi )}}}}$, (22)

I now pass on to calculate the total energy possessed by the ellipsoid when in motion along its axis of figure. In making the calculation I shall suppose that ${\displaystyle a^{2}>ab^{2}}$, i.e., that ${\displaystyle l^{2}}$ is positive. The case in which ${\displaystyle a^{2} can be deduced by the appropriate mathematical transformation.

I have shown {§ 22} that the total energy, viz. the volume integral of ${\displaystyle {\frac {\mathrm {K} \mathbf {E} ^{2}+\mu \mathbf {H} ^{2}}{8\pi }}}$, due to the motion of a charge on any surface, is

${\displaystyle \mathrm {W} ={\frac {1}{2}}q\mathbf {\Psi } _{0}+2\mathrm {T} }$,

where ${\displaystyle \Psi _{0}}$ is the value of the convection-potential at the surface of the body, and T is the magnetic part of the energy, viz., the volume integral of ${\displaystyle \mu H^{2}/8\pi }$.