164 Dr. J. W. ^Nicholson on the Bending of 



and % /t Joes not lead to finite oscillation in the manner of 

 <f)u in an exponential. This series becomes 



7 ,,=GK0)2r «(«*** -«.«*»>), . . . (84) 

 where 



" = MR,W^ (l + e 2l x*)/'2i, z{t h , v 2 )=<j>n-<i> nr ±m4>. 



Since Gr(#) annihilates the first harmonic, n=0 is taken as 

 the starting point instead of n—l. We restrict the investi- 

 gation for the present to points actually on the surface of the 

 sphere for this is the case to which controversy has hitherto 

 related and which moreover has a much simpler analysis. 

 Thus 



»=mR fl (l + <? l **)/2i, r(r l5 v 2 ) = ±<f>m. . (85) 



Recalling the formula of summation *, which in the present 

 case is very simple, since derivates of v higher than the first 

 are all zero, the scries 



S=2: o «Q^"(?) ..... (86) 

 has a sum 



s=* [" *.(u,+ 2j + u f + ^ ^ tm (87) 



where e = # i , , z , x = n/z 



^' •"^V'-'-l )« = ..;' 



In the present case, n is to be replaced by n + k, or m, so 

 that n Q is replaced by J. With the values of R„ and ^ 

 developed in an earlier section when n is not comparable 

 with s, 



"=t( 1+ W 



where zx—n-\-\. More accurate values of the functions are 

 found to introduce a correction which is only of order zaf/z 

 or A' 3 , and cannot affect the argument of this or of the next 

 section. There is no necessity to give the analysis of this 

 point, which can readily be supplied from the higher approxi- 

 mations already given. It will be necessary later to use 



* Messenger of Math. Oct. 1907. 



