384 ON THE ELECTRICAL CAPACITY OF [403
At this stage we may suppose I infinite in connexion with Hlt ffa, &c., so that the positive powers of r disappear. For brevity we write cos (nd - en) = Fn, and we replace r'1 by u. Thus
We have now to make <j> = faB,t the surface of the approximate cylinder, where fa is constant and
u = UQ + 8u = «0 (1 + GiGi + 02G2 + ...).
Herein Gn = cos (nd - <?„),
and the C"s are small constants. So far as has been proved, en might differ from en, but the approximate identity may be anticipated, and at any rate we may assume for trial that it exists and consider Gn to be the same as Fn, making
On the cylinder we have
-{- HQ
F2 + 8H&?Ft + . . . + ij? Op - 1) HvuJ>Fp}, . . .(4)
'O
and in this
8^ = 0^ + 0^ + 0^, + ...................... (5)
The electric charge Q, reckoned per unit length of the cylinder, is readily found from (2). We have, integrating round an enveloping cylinder of radius r,
and Q/0j is the capacity.
We now introduce the value of 8u/u0 from (5) into (4) and make successive approximations. The value of Hn is found by multiplication of (4) by Fn, where n=l, 2, 3, &c., and integration with respect to 6 between 0 and 2?r, when products such as F,F,, F2FS, &c., disappear. For the first step, where C2 is neglected, we have
(7)
Or
•Direct integration of (4) gives also fc = - #o log W)
(9)1)