An introduction to the mathematical theory of attraction ...

272 Systems of Conductors.
135. Efi'ect of Displacements son Energy.-If, be
a generalized coordinate on which the relative positions of
the conductors, A1, A, &c., depend, the charges e,, e,, &c., are
independent of F, but if 4 vary, so do the values of the
potential at the conductors as well as the coefficients pll, pl2
&c., qll, ql, &c.; then
dCWe d ( (Ip l - e 2 dP e2 22 2
Pli WV=2 + — e1.2 2 22 &c.)
d dW cvdF
d dV d[
The first part of the expression for, F Wv in which the
values V1, 72, &c., of the potential are not supposed to vary,
d y T V dentev bdWy
may be denoted by d; and substituting for dW d- V
dc tEd Vo dV' cdV '
&c., their values given by (6), Art. 129, we have
d d Wv. dV
When the charges are invariable from (5), Art. 129, we
have 2Î W = SeÎ V, and therefore if the values of the potential
and of W vary in consequence of a variation in [, we get
SW= -W r+ se-= dW V8+22W;
whence
dW. w r dJVFe
dtW = - 3W= d e. (13)
Equations (6) and (13) can be obtained directly from the
three expressions for Wby supposing e,, e,... e,, V, V'2.. V7,, and d all to vary; then,
2VFe + 2e y= 28TF= TWe, + WTV,
= d e d V dW dW )