Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/134

94 and the sum of the coefficients of induction of these conductors with respect to $$A_r$$ will be equal to $$q_{rr}$$ with its sign changed. But if $$A_r$$ is not completely surrounded by a conductor the arithmetical  sum of the coefficients of induction $$q_{rs}$$, &c. will be less than $$q_{rr}$$.

We have deduced these two theorems independently by means of electrical considerations. We may leave it to the mathematical student to determine whether one is a mathematical consequence  of the other.

Resultant Mechanical Force on any Conductor in terms of the Charges. 92.] Let $$\delta\phi$$ be any mechanical displacement of the conductor, and let $$\Phi$$ be the [sic] the component of the force tending to produce that  displacement, then $$\Phi\delta\phi$$ is the work done by the force during  the displacement. If this work is derived from the electrification of the system, then if $$Q$$ is the electric energy of the system,

If the bodies are insulated, the variation of $$Q$$ must be such that $$E_1, E_2$$, &c. remain constant. Substituting therefore for the values of the potentials, we have where the symbol of summation $$\Sigma$$ includes all terms of the form within the brackets, and $$r$$ and $$s$$ may each have any values from  1 to $$n$$. From this we find

as the expression for the component of the force which produces variation of the generalized coordinate $$\phi$$.

Resultant Mechanical Force in terms of the Potentials.

93.] The expression for $$\Phi$$ in terms of the charges is where in the summation $$r$$ and $$s$$ have each every value in succession from 1 to $$n$$.

Now $$E_r=\Sigma_1^t(V_tq_{rt})$$ where $$t$$ may have any value from 1 to $$n$$, so that