178 Lord Rayleigh on the Electrical Capacity of 



In the intervening space we may take for the potential in 

 terms of the usual polar coordinates 



<£ = H log {r/b)+B. 1 r- 1 cos (0-eO + K 1 r cos (0 — e/) + . . 



+ R n r- n cos (n0 — e n ) +~K n r n GOs(nd — e n '). 

 Since (£ = when r=6, 



e n ' = e ni K n =-E n b- 2 «, 

 and 



^ = Holog(r/6)+H ] (^_^co S (^-e 1 ) 



+ H 2 (~ 2 -g)cos(20-e 2 )+ (1) 



At this stage we may suppose b infinite in connexion with 

 H 1? H 2 , &c, so that the positive powers of r disappear. 

 For brevity we write cos (n# — e n )=¥ n , and we replace r~ l 

 by u. Thus 



<£ = -H log 06) +H 1 wF 1 + H 2 w 2 F 2 -h (2) 



We have now to make <£ = (/>i at the surface of the 

 approximate cylinder, where <p 1 is constant and 



u=u + 8u=u {l + C 1 G l + G 2 G 2 + . . .)• 

 Herein 



Gr»= cos (n0—e n ), 



and the C's are small constants. So far as has been proved, 

 e n might differ from e», but the approximate identity may be 

 anticipated, and at any rate we may assume for trial that it 

 exists and consider Gc n to be the same as Y n , making 



u=u + 8u=u {l + C 1 F 1 + C 2 F 2 + . . .). . . . (3) 



On the cylinder we have 



</>! = - H log (uj>) + Hi^Fi + IWF 2 + . . . 



+ - \ - H + K 1 u ¥ 1 + 2H 2 u 2 F 2 + 3H 3 VF 3 + . . .1 



+ Q 2 {iH + H 2 u 2 F 2 + 3H 3 VF 3 + . . . 



+ ip(p-l)H p ttfF P },. . . (4) 

 and in this 



8u/u =C 1 F l + O f F f + C l F,+ (5) 



