Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/135

93.] Now the coefficients of potential are connected with those of induction by n equations of the form

and $$\tfrac{1}{2}n(n-1)$$ of the form

Differentiating with respect to $$\phi$$ we get $$\tfrac{1}{2}n(n + 1)$$ equations of the form where $$a$$ and $$b$$ may be the same or different.

Hence, putting $$a$$ and $$b$$ equal to $$r$$ and $$s$$, but \Sigma_s(E_s p_{rs})=V_r, so that we may write where $$r$$ and $$t$$ may have each every value in succession from 1 to $$n$$. This expression gives the resultant force in terms of the potentials.

If each conductor is connected with a battery or other contrivance by which its potential is maintained constant during the displacement, then this expression is simply

under the condition that all the potentials are constant.

The work done in this case during the displacement $$\delta \phi$$ is $$\Phi \delta \phi$$, and the electrical energy of the system of conductors is increased  by $$\delta Q$$; hence the energy spent by the batteries during the displacement is It appears from Art. 92, that the resultant force $$\Phi$$ is equal to $$-\tfrac{dQ}{d\phi}$$, under the condition that the charges of the conductors are constant. It is also, by Art. 93, equal to $$\tfrac{dQ}{d\phi}$$, under the condition that the potentials of the conductors are constant. If the conductors are insulated, they tend to move so that their energy  is diminished, and the work done by the electrical forces during  the displacement is equal to the diminution of energy.

If the conductors are connected with batteries, so that their