386 ON THE ELECTRICAL CAPACITY OF [403
by which Hn is determined by means of definite integrals of the form
FnFpFqdO, .............................. (17)
n, p, q being positive integers. It will be convenient to denote the integral on the right of (16) by In, In being of the second order in the (7's.
Again, by direct integration of (4) with retention of C3,
rJfi
— (C1Fl + Ct
CF
In the last integral we may substitute the first approximate value of Hp from (8). Thus in extension of (11)
+ %p (p -1) GpFp}.............(18)
The additional integrals required in (18) are of the same form (17) as those needed for In.
As regards the integral (17), it may be written
Now four times the latter integral is equal to the sum of integrals of cosines of (n — p — q) 8, (n — p + q) 6, (n +p — q) 6, and (n +p + q) 6, of which the last vanishes in all cases. We infer that (17) vanishes unless one of the three quantities n, p, q is equal to the sum of the other two. In the excepted cases
K .................................. (19)
If p and q are equal, (17) vanishes unless n = 2p; also whenever n, p, q are all odd.
We may consider especially the case in which only Cp occurs, so that
u = Uo(l + Cpcospd) ......................... (20)
rzv a In (16) IB = (2p+l)<being symmetrical about the line 0 = 0. Thus from (4), as an extension of (7) with inclusion of O2,