238 Dr. J. W. Nicholson on the Asymptotic 



The value of S, in the second asymptotic substitution, may 

 now be expressed, but this substitution is of small importance, 

 as it is identical with the third in all useful cases. 



Again, by (44), if n<i 



v J 2 _ n (z)=rJ- Zn (2zdn0)d6. . . . (45) 



Thus, if the argument of the functions be constantly z 

 when not expressed, 



2tt J n |^ 1 = j^ ^ J 2n (2z sin 6) d$ 



~dn 



2irJ.M^= i ' £ J_ 2n (2zsmd) dO. 



Thus 



or, when n is made integral, its only possible value being 

 zero, 



7rJ (z)Y Q (z)=CY (2zsme)dO. . . (46) 

 h 



A more general result is proved in the next section. 



The Formula for J n {z) Y n (z). 



An expression for the product of two Bessel functions of 

 different types, when n is integral, has been given by 

 Neumann *. But it is somewhat unsuitable for our purpose, 

 and an alternative is now developed which can, however, be 

 formally identified with that of Neumann. 



By (43), the argument being z unless otherwise specified, 



But 



27rJ n ^"= f ^ J 

 O n Jo 



"dn 



J ,7r 

 cos (w sin <f) — 2n<j)) d<f> 



— sin 2mr 1 d<j> e " 2n( P ~ w sinh t 

 Jo 



* Bes&eliche Functionen, p. 65. 



