182 Lord Rayleigh on the Electrical Capacity of 



we are supposing not to occur. Also the last integral in 

 (18) makes no contribution, reducing to 



T 27r JP) 

 \p{p-V)Ov z \ ~cos*pd, 

 Jo ^ 

 which vanishes. Thus 



^/Q=21og(« J)-(p + i)CV, . . . . (21) 



the same as in the former approximation, as indeed might 

 have been anticipated, since a change in the sign o£ C p amounts 

 only to a shift in the direction from which 6 is measured. 



The corresponding problem for the approximate sphere, 

 to which we now proceed, is simpler in some respects, though 

 not in others. In the general case u, or r" 1 , is a function of 

 the two angular polar coordinates 0, co, and the expansion 

 of cu is in Laplace's functions. When there is symmetry 

 about the axis, co disappears and the expansion involves 

 merely the Legendre functions Fn(/n), in which /a=cos#. 

 Then 



M = Wo + ^= Mo {l + C 1 P 1 (/.) + C 2 P 2 (/ A )+...}, . . (22) 



where Ci, C 2 , . . . are to be regarded as small. We will 

 assume Su to be of this form, though the restriction to 

 symmetry makes no practical difference in the solution so 

 far as the second order of small quantities. 



For the form of the potential (<fi) outside the surface, 

 we have 



(/)=H ^ + H ] ^P 1 (^) + H 2W 3 P 2 (/x)+ . . .; .... (23) 



and on the surface 



^ 1 = H oMo + H 1 VPi + H 2 VP 2 + . . . 



+ $u{H + 2u H 1 P 1 + 3VH 8 P 8 + . . .} 



+ (6^) 2 {H 1 P 1 + 3u H 2 P 2 + . . . + ip(p + l)u P-mpP P \, (24) 



in which we are to substitute the values of Su, (8u) 2 from 

 (22). In this equation fa is constant, and H l5 H 2 , . . . are 

 small in comparison with H . 



The procedure corresponds closely w T ith that already 

 adopted for the cylinder. We multiply (24) by P„, where 

 n is a positive integer, and integrate with respect to n over 

 angular space, i.e. between —1 and 4-1. Thus, omitting 

 the terms of the second order, we get 



Wo "H n = -H C. ...... (25) 



as a first approximation to the value of H*. 



