704 Dr. J. W. Nicholson on the Asymptotic 



Second form for the function R. 



The formula just proved suggests the existence of an 

 asymptotic series of the form 



f = h + h + H >+ , • . . . (25) 



2z x x 6 x b 



where x = (4:Z 2 — m 2 )* and X^l. 



Such a series may be obtained directly from the differential 

 equation. Writing U = 2zy in (8), it is found that 



, r + 3 ^ + 4cy(l- m ^ 1 )-f4y=0, . . (26) 



where 4n.n+l has been written as 4m 2 — 1. With a new 

 independent variable x = (£z 2 — m?)*, this may be reduced to 



^ 2 ( t u 2 + m 2 )V'' + 3^(^ 4 -m%'' + 3mVH-«r 4 (l + ^ 2 )y + ^ = 



.... (27) 



Writing Xi \ 9 



* y= — +-1+ 



u x x i 



the relation between successive coefficients (s odd) becomes 

 (s + 3)\ s+ 4+(s + 2f\ s+2 +2mh.s+l.s + 2.\ s 



+ m 4 s.s-2.s + 2.\ s .2=0, . . (28) 



and corresponding to Ai = l, 



X.= -i, Xi=ft27-24m«) 



A 7 = 1 1 g(1160m 2 -1125-4077i 4 ). . . (29) 



Any succeeding coefficient may be at once found by (28). 

 Every third term in the resulting series is two orders (in m 

 or x, and therefore in n or z) smaller than that immediately 

 preceding. Thus the second is two orders smaller than the 

 first, the fifth two smaller than the fourth, and so on. 



Finally, 



R 1 1 27 -24m 2 



2z (4~ 2 -m 2 )* 2.(4,^ 2 -m 2 )l ' 8.(4^-m 2 )f 



1160 m 2 - 11 25 -40m 4 

 16(4^ 2 -m 2 ) 



+ - "T^Tr^r + (30) 



The terms here written give the value of R correctly to 

 four places of decimals when z is only 10, and n = 8, a case 

 in which n and z are nearly equal. 



