194 
DR. T. J. FA. BROMWICH ON THE 
Provided that 2 — (n + -|-) is of an order higher than the formulae are 
S„ (2) = P sin \p-, 
■«//• = 2 sin a + |- 7 r— (n + |-) a, 
r E„ (2) = Re-‘^ 
I where 
(6‘l) ^ R* = l/sin a, 
and 
cos a = {n + ^)lz. 
We shall need also the corresponding formulae for (2) and E'„ (2); it will be seen 
that 
E' (2) 1 dR d-dr 1 /1 cos a da. 
E„ (2) R dz 
because ( 3 ‘86) gives 
Hence 
dz 
R2^ = 1. 
dz 
(6-2) 
where* 
F4(2) _ \ s_ L 
E„(2) i)2(tanx + 0 
, 1 cos a da 1 cos^ a 
tanx = i— 
sin a dz z sim a 
Similarly, we find that 
/c-q\ S'„(2) 1 c^R , ddr 1 / 4, , 4. I \ 1 cos(i^ + x) 
o / \ = (-tanx + cot — Y -— 
S„ (2) R d2 ^ dz ^ ^ ^ R^ sin Y cos x 
The formulae to be used finally are those for A„ and C„, given in ( 471 ), thus 
we take 
A 
(2) ’ 
C = - 
S. (^) 
where 2 now denotes /ca. It follows from (6‘l) above that 
C„ = -sinV^e+‘'^ = +|-t(e^‘^-l), 
and using (6’ 2 ) and (6’ 3 ) we see that 
A„ = cos(V- + x) = + 
It is now an easy matter to write down an approximation to the functions M and 
N defined in ( 4 ’ 8 ), provided that 6 is not near to 0 or tt. Under these conditions we 
can take the approximate value 
P„ (cos 0 ) = ^ I -?r—) COS {(n + i) O-ivr}, 
V \7lir Sin 0 / 
* Under our conditions a is not near to zero and z is large, so that x is small. 
