Expansions of Bessel Functions. 241 



Moreover, if identically 



1+^+^*+ ... = (L + \ 2 tf 2 -fM' 4 4- ... )"\ (54) 

 Then 



- - -f^ 4 tan « - £j (tan « - 1 tan V) 



+ ^(tana— A-tan :j a+l - tan 5 a) — ... j>, (55) 



and the second, fifth, &c. terms in the large brackets are 

 each two orders smaller than the preceding. 

 An identical relation 



"•-S+Sf---* • • • • (56) 



proved later *, has been used in the reduction. 



R may be arranged more conveniently for some purposes 

 in the form 



z tfif z W 10; 2 $&< z h x l* 

 3 , 2 2 + 5 I 2 4 + 6 S 2 1 + 7 ! 2« 



56* 2 W '280^W, \ i ~ 7 . 



+ Tf 2 6 " + ~TT 2 6 + ••'J v^-n*' ^°' ; 



where $i=d/dr, S 2 =B/dw, 



being here given to an order z~ 6 when s and n are of the 

 same order. We shall refer to this system of expansions 

 subsequently as (A). 



The Remainder in the Expansion o/R. 



From the previous paper the remainder after r terms in 

 the expansion of R is 



where 



v= sinhz + fit, v'= cosht + /u, \=2,scoshi^, \fi — 2n, 



and S denotes an addition of the two values corresponding 

 to the ambiguity. 



Now it is well known that the integral 



R-*{l+. : 



= j>> 



e-™dt, .... (59) 



* Cf. (67) infra. 

 Phil. Mag. S. 6. Vol. 19. No. 110. Feb. 1910. R 



