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



The notation has been slightly changed in the last formula. 

 Moreover, x being ra/s, where m = n + %, and the accent 

 denoting "dfdx, i\ is never zero within the range of variation 

 of m, and v 2 f is zero at a certain " zero point " given by 



^=sin</>/(l-2ccos(/)H-c 2 )i, c = a/r. . (113) 



These results are exact. If, on the other hand, it he 

 possible io replace the zonal harmonic by its asymptotic 

 expansion throughout the series, the zero point is given by 

 x == r sin Q\ R, or 



#=sin<9/(l-2ccos04-e 2 )§. . . . (114) 



These results are to be found in the earlier section headed 

 "Vanishing of the derivate of an exponent/' The problem 

 in hand is the determination of a second approximation to 

 the sum of a series of this type, where the exponent has only 

 one zero point. The summation may be replaced at once by 

 an equivalent integral as before, and the main difficulty is 

 therefore the evaluation of this integral to a higher order. 



Let S be a series of the type customary to this paper, 

 given by 



S = 1 ue lz \ (115) 



n=0 



where * is large, and u and v are expressed as functions of 

 0:sc(n + £)/?. Then by the previous summation formula 



u, . u, 



s=*{ «k(u.+ -£ + ^'+...) 



where U , U l5 ... are given in slightly different notation by 

 (34) and e=l/2z corresponding to ?i = 0. As was indicated 

 previously, the harmonic term of zero order in 7 is zero, so 

 that rc==0 may be taken as starting point instead of w = l. 

 Writing 



U + -^+ ...=U (116) 



Then 



S^zC dx^e lzv } .... (117) 



and it is to be assumed that v' = at one, and only one point 

 in the range, namely, v = v corresponding to x = x . More- 

 over, U is not oscillatory. In view of defmiteness, v " will 

 be given a positive, and v a negative sign. This has already 

 been seen to be the case in the series most important in the 



