236 Dr. J. W. Nicholson on the Asymptotic 



Then, by direct multiplication, an ascending series is 

 obtained whose leading terms are 



Ma=z; + o 



>2n^2 2n.n 2 -l-' 



and which, moreover, must satisfy the associated equation, 

 Thus by comparison of series 



1*2 = 2^, 



and 



r»7r 



Tl= l— cos2*» rj sin2nA,J o( 2 *sinA-)<fo. • (38) 



This result was known to Lorenz for the case in which 2n 

 is an odd integer. The substitution (36) is more convenient 

 than (9) when n is non- integral and greater than z. Thus 

 for all real values of z and n, the latter not being an integer, 



J n {z)J_ n {z) = ~ — f sin 2nx . J (2* sin x) dx. (39) 



When n is an integer, evaluating the form then presented, by 

 an^obviously legitimate process, 



( — ) ?l J« W = — =r~ I sin 2nx . J (2z sin x) dx -f- s - sin * ?7r 



( — )" f * 

 = v — ;,-- I 2x cos 2n# J n (2r sin x) dx 



* Z Jo 



( V* C' 7 * 

 — - — f- — j 2 cos 2nx J (2z sin #) <7#, 

 7r " " Jo 



or 



1 fir 



Jfj(^) = " 1 sin 2na . J (2z sin <r) d#, n = integer. (40) 

 Jo 



This follows otherwise from a result given by Neumann*. 

 The determination of T (T i being infinite) when n is integral 

 is somewhat difficult, for u 2 and u 3 cease to be distinct. For 

 this determination a more direct method is useful f. 



It was shown that if t be defined by the third asymptotic 

 substitution, 



* Cf. Gray and Matthews, p. 28. 



f Cf. the formula for 3n{z)Y ? i(z), infra. 



