240 Dr. J. W. Nicholson on the Asymptotic 



The value of T when n is integral may now be deduced, 

 and thence t by (41). 



Expansions of the First Type. 



Beturning to the value of R given in (31), it is first to be 

 noticed that the evaluation (asymptotic) of the integral, given 

 in a previous paper *, was in no way dependent upon the 

 restriction of 2n to an odd integer. Accordingly, this 

 evaluation may be used in the general case. The same 

 applies to the subsequent treatment of p as given by (35) of 

 the present paper. 



Thus quoting the values of R and p previously obtained, 

 we obtain the following asymptotic expansions when n is less 

 than z, and z—n is not very small : — 



When n is not an integer 



/2R\* 



J_«(~) — COS 717T J;i (z) — I — - 1 COS p Sill W7T, 



'2R\* 



and when n is an integer, 



Y„(c) = - r-y-Y cos p, 



where if n = z sin a, defining an angle a, 



R ■ = sec a + ^fsec 3 a-f 4 4 sec 4 a+ ... , . . (52) 



where 



1 27-96// 2 _ 4640n 2 -1125-640n 4 



^2 — — 2"3 y A i 2 7 ' 6 — 2*° ~~ ' 



and 



4.* + 3.\,+B+(*+2) 8 A w+ i + 2n 2 *.*+.l.*+2 .\ 5 _i 



+ A.s 2 -4.X,_ 3 =0, . . (53) 



and every third term of R, beginning with the second, is 

 two orders (in z or n) smaller than those before. 



* Phil. Mag. Dec. 1907. 



