246 Dr. J. W. Nicholson on the Asymptotic 



and the second, fifth, ... terms are each two orders in n smaller 

 than those preceding, and 



/ = »(ta„h/3- / 3)-ilog e 2-i 3 (^-|^ + i§...) 



+ ^coth/J- £>th s j8H ...) + ?in _ B 2 2! + B34!_ (?2) 



n$x* 3ti 2 / n 2 ! ?t 3 4 1 »* b I 



where 1, /i 2 w -2 , a^ -4 , decrease in order by w~ 2 at each third 

 member of the series. 



This will be called expansion B. 



This value of J»(s) continues to hold when n is an integer, 



and it may be expected that Y„(Y) will then replace 



{ J - n (z)— cos mrJ n (z)\ cosec nir ; but the formal proof is 

 necessary owing to the use of T x in place of T. This proof is 

 given in the next section. 



Expansions when n is integral and greater than z. 

 In order to verify the result last suggested, concerning the 

 expression for Y n (z), it is only necessary to prove that if 



J*)- (g)v. 



Then the value of T is, to a first order, given by 



For if this be true, the subsequent proof follows the lines 

 above. Now by (50) 



Jn(z)Y (z)^i Tcos 2nx . B (2z sin x)dre- - ( °\ty {" d0 e -»*-*H*f*** 



Jo ^Jo Jo 



=*Ii-|l,(Bay). p . . . (74) 



The first approximation to I u found in the usual manner, 

 is, when n > z, w is large, and not too close to z in value, 



I = ir( i L W 



7rJo v&— -zsin$ w + £sin<£/ ^ 

 _4^( l d^_ 



7T 



\/V 2 — Z 



..sin- 1 ' (75) 



