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



Eestating the result in simpler notation, it has been shown 

 that if a series dependent on the lower limit is negligible in 

 comparison, the second approximation to the integral 



S=5 f MC .... (130) 



where z is large, and v has a single minimum in the range, 

 and no other turning point, U being finite and non^oscillatory 

 everywhere, is given by 



S = 2(«)»JV,^2}«*'*?, . . . (131) 



where, all values of functions relating to the zero point 

 where the minimum occurs, and suffixes or accents on the 

 right denoting differentiations at the zero point with respect 



to It!, 



V =U(2f 2 )-i, 



Application to Series Summation. 



The sum of a series of the usual type, with our customary 

 notation, 



?t=o \ ^ / 



is 



'here 



z 

 and by (34), 



U 2 = iu"ir 2 ^iL{v"u'^lv" f u)f z -iuv" 2 ^ , (133) 

 where 



so that 



f;=^"^+i, (134) 



the accent denoting differentiation with respect to #. Let 

 us suppose that the significant zero points can be reduced 

 to one. Then the sum is given to a second approximation 



