184 Lord Rayleigh on the Electrical Capacity of 



difference. I£ we take 



g w / Wo =F 1 +F 2 -f . .. + F„, .... (32) 



where the F's are Laplace's functions, 



1 (V 2 



j— I j F p 2 d/jb dco corresponds to v . 



This substitution suffices to generalize (30), (31), and the 

 result is in harmony with that formerly given. 



The expression for the capacity (30) may be tested on 

 the case of the planetary ellipsoid of revolution for which 

 the solution is known *. Here () 2 = ^e 2 , e being the eccen- 

 tricity. It must be remembered that a in (30) is not the 

 semi-axis major, but the spherical radius of equal volume. 

 In terms of the semi-axis major (a) , the accurate value of 

 the capacity is a^/sin -1 ^. 



We may now proceed to include the terms of the next 

 order in C. The extension of (25) is 



« »H„/H = -C» + |(2n + 1) f ^,,{0^ + . . . + C P P,} 



{2C 1 P 1 +...+ ( ? +l)C,P,},. (33) 



where in the small terra the approximate value of H„ from 

 (25) has been substituted. We set 



r +i d>p„{c 1 p 1 +...+c J ,p P t 



t2C 1 P 1 + ... + (g+l)C,P 4 }=J n , . (34) 



where J n is of order C s and depends upon definite integrals 

 of the form 



£' 



?n?p? q d^ (35) 



n, p, q being positive integers. 



In like manner the extension of (26) is 



&/CK=i-§(v-fcy- • • • - g^tv 



+ i{2C 1 J 1 + 3C 2 J 2 + 4C 3 J 3 + ...} 



-ir +1 ^(CiP, + C 2 P 2 + . . .) 2 {CiPi + 3C 2 P 2 + . . . 



+ ip(p + l)CP,} (36) 



Here, again, the definite integrals required are of the 

 form (35). 



* Maxwell's ' Electricity/ § 152. 



