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



force is given as in (112) by 



?/ = im(R, l R ar ) 2 (l + <r lx »), ( Vl , v 2 ) = ( f in -,^ nr j : m(j), . (158) 

 and 



g ^ )c ° ka 2 7r d/ij x/2 Vcos<£-cos0' 



and where the series with exponent v x may be neglected, to 

 the order concerned, because the derivate of this exponent 

 cannot vanish. The derivate of the other exponent vanishes 

 where 



x = sin 0/ ( 1 — 2c cos cf) + c 2 )*, c = a/r. 



We shall restrict attention at once to the effect at some 

 distance from the sphere, so that c is zero, and the zero 

 point is A' = siii(£. The reduction of the formulae is simple, 

 by virtue of the theorem given in (138), that the sum of 

 a series 



% = ^ue l " v (159) 



with the usual notation of this paper, is, under certain con- 

 ditions already shown to be satisfied, given to a second 

 approximation by 



£iy*+*r m m (16Q) 



H^K^i\ 



the values being taken at the zero point, and suffixes 

 denoting differentiations with respect to x or (n + ^)/z. 

 The magnitude //. is given by 



^rv'iv^?- • • (101) 



In the neighbourhood of the zero point, <f> n and cf> nr become 

 as usual, the latter relating to a large value of r, 



<j>n=-7+~] Vl-^ + ^sm *£— — V 



L " . . (162) 



These are the first approximations. As in (147), we may 

 continue to use these in finding a second order solution 



