Page:Philosophical magazine 23 series 4.djvu/36

20 Prof. Maxwell an the Theory of Molecular Vortices

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) that

and we know by (105) that

whence

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

or by (115),

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

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

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

And if $$e_1$$ is confined to a small body,