Page:Philosophical magazine 23 series 4.djvu/36

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20 Prof. Maxwell an the Theory of Molecular Vortices

U=-\Sigma {\frac {1}{2}}(Pf+Qg+Rh)\delta V,

(116)

where P, Q, R are the forces, and f, g, h the displacements. Now when there is no motion of the bodies or alteration of forces, it appears from equations (77)^[1] that

P=-{\frac {d\Psi }{dx}},\ Q=-{\frac {d\Psi }{dy}},\ R=-{\frac {d\Psi }{dz}};

(118)

and we know by (105) that

P=-4\pi E^{2}f,\ Q=-4\pi E^{2}g,\ R=-4\pi E^{2}h;

(119)

whence

U={\frac {1}{8\pi E^{2}}}\Sigma \left(\left({\frac {d\Psi }{dx}}\right)^{2}+\left({\frac {d\Psi }{dy}}\right)^{2}+\left({\frac {d\Psi }{dz}}\right)^{2}\right)\delta V.

(120)

Integrating by parts throughout all space, and remembering that $\Psi$ vanishes at an infinite distance,

U=-{\frac {1}{8\pi E^{2}}}\Sigma \Psi \left({\frac {d^{2}\Psi }{dx^{2}}}+{\frac {d^{2}\Psi }{dy^{2}}}+{\frac {d^{2}\Psi }{dz^{2}}}\right)\delta V.

(121)

or by (115),

U={\frac {1}{2}}\Sigma (\Psi e)\delta V.

(122)

Now let there be two electrified bodies, and let $e_{1}$ be the distribution of electricity in the first, and $\Psi _{1}$ the electric tension due to it, and let

e_{1}={\frac {1}{4\pi E^{2}}}\left({\frac {d^{2}\Psi _{1}}{dx^{2}}}+{\frac {d^{2}\Psi _{1}}{dy^{2}}}+{\frac {d^{2}\Psi _{1}}{dz^{2}}}\right)

(123)

Let $e_{2}$ be the distribution of electricity in the second body, and $\Psi _{2}$ the tension due to it; then the whole tension at any point will be $\Psi _{1}+\Psi _{2}$ , and the expansion for U will become

U={\frac {1}{2}}\Sigma \left(\Psi _{1}e_{1}+\Psi _{2}e_{2}+\Psi _{1}e_{2}+\Psi _{2}e_{1}\right)\delta V.

(124)

Let the body whose electricity is $e_{1}$ be moved in any way, the electricity moving along with the body, then since the distribution of tension $\Psi _{1}$ moves with the body, the value of $\Psi _{1}e_{1}$ remains the same.

$\Psi _{2}e_{2}$ also remains the same; and Green has shown (Essay on Electricity, p. 10) that $\Psi _{1}e_{2}=\Psi _{2}e_{1}$ , so that the work done by moving the body against electric forces