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

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 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,

$\Phi \,\delta \phi +\delta Q=0\,$,

(3)

or ${\color {White}xxxxxx}\Phi =-{\frac {\delta Q}{\delta \Phi }}$.

(4)

Here

$Q={\tfrac {1}{2}}(E_{1}V_{1}+E_{2}V_{2}+\And \!\!{\mbox{c.}})$

(5)

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

$Q={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}p_{rs})$,

(6)

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

$\Phi =-{\frac {dQ}{d\phi }}=-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})$

(7)

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

$\Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}(E_{r}E_{s}{\frac {dp_{rs}}{d\phi }})$

(8)

where in the summation $r$ and $s$ have each every value in succession from 1 to $n$.

Now $E_{r}=\Sigma _{1}^{t}(V_{t}q_{rt})$ where $t$ may have any value from 1 to $n$, so that