170 Dr. J. W. Nicholson on the Bending of 



Thus again defining G(0) by the operation 



G(0) . tv = 2t ^ 6 d f * !^_ (m 



hd 2 ir cWJ 9 • ;V(cos0-cosft) # W 



Then for points on the surface, by (83), 

 yp = G(0) f niRn(l + e 2 ^n)smm6 



= -2tG(0) %2, v l^.msmmd), 



and as m sin mft/m 2 — v 2 is an even function of m, the 

 summation may be replaced by half that from — co to oo . 

 Therefore 



y P = - l G(0)X,2va v 2 . / , • • • (95) 



where 2wi takes all possible integral values, positive or 

 negative. But the last summation may be effected. For 

 consider the function f(y) sec 7rv, where /(?') has no poles. 

 The poles of the function are given by cos7n>.= or 



TTV = ± (/) + i)tt = + W7T, 



where m is a typical value of m in the desired summation. 

 The residues at these poles are f(±m)JTrsm nm, so that 



/"(v)soc7rv = 2, • . ('^ — '- _i-- v ' ) 



_oc 7TSni 7><7T\V— //I V+/>/ /' 



w taking half integral values. Identifying this with 

 the series 



^ m sin mft 



4_1^ /sin mft sin ?«ft 



we find at once that 



^ m sin »<ft _ 7rcosv(7r— ft) , qfi s 



-« m 2 — I/ 2 COSV7T ' - ' • • \ J 



and therefore 



7P = — 27H. 2„va„G (0) . sec vir cos v(7r — ft) . 



Assuming the result of the next section, that the first pole 

 is one for which v = s—idj3, where /3 is an ordinary 

 numerical quantity, we have, ft being less than ir throughout 

 the operation of G(0), 



cos 1/(77— ft) = e tv ^-V = ^ _„, 



COSV7T ,,<*■'' 



