dO _ 

 sin0 



Expansions of Bessel Functions. 247 



To the same order, 



I 2 =- I ' ( W. \{2n + 2z sin 6 cosh £)-i*-s»* -2.* •.-»;• 3iah*1 



= - i - 4 ^ —sin -1 - [ , on reduction, . . . (76) 



\/n*— z* y- - n ' 



Thus 



giving the proper value for T. 



Finally, in expansion B, when n is integral, we write 



J, lW = (g)V, Y„(;)=-(^;. . (78) 



and the functions (T, are those previously given. 



The limits of accuracy of expansion B will be the same as 

 those for A, with 11 and z interchanged. 



The transition between different forms of expansion. 



It will now be shown that there is no range of large values 

 of n or z for which expansions are yot to be determined, when 

 n and z are real, and that each expansion passes naturally 

 into the other, without the necessity of an intermediate 

 expansion. We shall first define expansion as that of a 

 previous paper, where n—z is not largo in comparison 

 with ~i, viz. 



3Jn(z) — -(r) { fi(p) cos »«"+/s(p) +Mp) sin »«•}. » not integral. 



Y„(:) =_Q J {/;(P) + /3(P)},» integral (79) 



