194 DR. T. J. I'A. BROMWICH ON THE 



Provided that z (n + %) is of an order higher than z 1 '', the formulae are 



| where 



(61) R*=l/sina, ty = z sin a+fa- (n + %) a, 



and 



cos a = 



We shall need also the corresponding formulae for S' n (z) and E' n (z) ; it will be seen 

 that 



E' B (z) _ \_ dE, _ d\ff J_ / 1_ cos ofo. \ 



ET == R dz " ' ~fa '' R 2 \ 2 sin 2 a cZz /' 



because (3'86) gives 



dz 

 Hence 



F/. (z) 1 



(6-2) 



where* 



i cos a da. , cos 2 a 

 tan x = f-T-s- -y- = 2 ^i 



sin 2 a dz z sin 3 a. 



Similarly, we find that 



/>.o\ 

 (6 3) 



S' B () 1 cZR , rZ\/r 1 



" = - - i 



- ^- =^^ 



S n (z) R dz cZz R 2 v R 2 sin ^ cos x 



The formulae to be used finally are those for A n and C n , given in (4 71), thus 

 we take 



where z now denotes m. It follows from (G'l) above that 



and using (6 '2) and (6'3) we see that 

 A = -le' 



It is now an easy matter to write down an approximation to the functions M and 

 N defined in (4'8), provided that is not near to or TT. Under these conditions we 

 can take the approximate value 



P B (cos 9) = /v / 



cos 



\it7T olll {// 



Under our conditions a is not near to zero and z is large, so that x is small. 



