702 



Dr. J. W. Nicholson on the Asymptotic 



R to a high order, as in the usual theory of asymptotic ex- 

 pansions. The series is evanescent at the upper limit on 

 account of the exponential, and thus, if r =0, 



If E denote the operation ^r == -/ 



dv ~ v' dt ' 



=i( 



l+~° + 



E 2 



+ ... - 



)h- 



(17) 

 (18) 



X ' A, 2 / Vq 



which may be symbolically written 



I=Ck--E )-K(v ')-\ 

 where the sign of equality denotes asymptotic equivalence. 



Now R = — ( °° (li + L) d^, .... (19) 



77 Jo 



where (I 1; I 2 ) = g-3*siiiii*cosh*±mt^ 



Jo 



If m is of the same order as ~, and such that z — m is not 

 of low order, these integrals are of the form treated above. 



Writing \ = 2 c cosh yjr, m = \fi, P<1> 



and denoting I x or I 2 by I, 



1= I e-W^tTrtdt. .... (20) 

 Jo 



Thus in the above notation, 



v = sinh t + />tf, i' = 0, v ; = 1 + //,, 



By help of these results it is readilv proved that, if w = v \ 



Ei(i)=-i 



,4/1) 1 10 



°\w/ ' w b ' W 



(21) 





+ 



1 ^_^° 



and so on, where w = 1 + At, Aw = 2c cosh yfr-t-m. 



