388 ON THE ELECTRICAL CAPACITY OF [403
Direct integration of (24) gives dfi+u,
or on substitution for fTn from (25)
......... (26)
f+i 2
inasmuch as P/ (^) C?/A = g--— ......................... (27)
J -1 *P + -!• .
As appears from (23), Hn is identical with the electric charge upon the sphere, which we ' may denote by Q, and Q/<j5i is the electrostatic capacity, so that to this order of approximation
Capacity = u^ {l + fC/+ . . . + ^ (7/1 ............ (28)
I ^P + *- )
Here, again, we must remember that u^1 differs from the radius of the true sphere whose volume is equal to that of the approximate sphere under consideration. If that radius be called a
and Capacity = al+ ~ + ... + ~_ c , ....... • ..... (30)
in which C'a does not appear.
The potential energy of the charge is ^Q2 — Capacity. Reckoned fron: the initial configuration (G= 0), it is
5
It has already been remarked that to this order of approximation th< restriction to symmetry makes little difference. If we take
.....................(32)
where the Fs are Laplace's functions,
Iff C2
-.— FJ* dfidoi corresponds to =•—• _ . 47T J J x 2>p -f-1
This substitution suffices to generalize (30), (31), and the result is in harmon with that formerly given.
The expression for the capacity (30) may be tested on the case of th planetary ellipsoid of revolution for which the solution is known*. Her
* Maxwell's Electricity, § 151. order, we get