176 Miss Dorothy Wrinch on a Generalized 



The values o£ the other relevant r a s coefficients found by 

 induction are as follows : 



v , (n-l)(n -2)^ , n(n-l)(n-2)(3n-5) 



^ | ( w -2)(n-3) s t ( n-l)(n-2)(n-3)(3n-8) s 



24 i 



| ^_l)( s _2)( 5 -3) 2 (3 5 -8) 

 n(n-l)(»-2),, ~ 



2 «2=^ i +(»-i)2. 



^"7 ' 2 7' 24 



+ 3> 



«! and « 3 are then found to be 



lVs^lj [l% 2 + (s«l — 2« 2 )l«2 + 2«l • l«s]/2 2 «l 2 , 



respectively. Subsequent coefficients « 3 , a 4 ... can be found 

 in the same way, but they will be increasingly cumbrous. 

 They are not, however, of the same practical importance, as 

 a x and a 2 will, in general, alone be required in applications. 

 The form of the asymptotic expansion of F n _i(V) is then 



where 



"- 1 n — 1 



S n = 2, a s -\-- 



2 



and n z r represents an nth root of z, the summation being- 

 taken over the nth roots of z. 



Now, in the case of a value of n z r which has a negative 

 real part, whatever the value of f r (n) the corresponding part 

 of the asymptotic expansion, namely 



i--r ■«/ r »«""- > ( i +s T 7^A 



is asymptotically equivalent to zero. Hence, although it is 

 true that 



.F.. 1 (z)^|*r^s/»«'*(i+J 1 -^i) • a) 



f r (n) corresponding to a value of n z r with a negative real 

 part, can take any value whatever, and there is therefore 



