SCATTEEING OF PLANE ELECTRIC AVAVES BY SPHERES. 
195 
giving a corresponding approximation 
Thus 
where 
F. (cos e) = 2!^' Bin{(yi)9-M 
sm 0 ^(2?^7^sln0) 
sin 9 A„P'„ (cos 0) = — 
n (n+1) 
x/{2n7r sin 0) 
ft) = (w + |-) 0—^-TT, 
and we have replaced + (^+l)} by unity, because n is large. 
Thus, on putting ;,(■ = 0, we find that (4’8) gives the approximation 
(6-4) 
M = —cos (p 
kv v^(2n7rsin0) 
Similarly, we get the formula 
(6-5) 
N = + sin 0 
kv ^(2w7rsin0) 
With series of this type, the leading part is found by making the index of the 
exponential stationary (regarded as a function of n). Now in both M and N there 
is one index only, cr = 2i/r + n7r—<«), which can be stationary : and the condition is 
Now, from (6'l) 
d\fj- 
an an 
da 
= {z cos a —(n + |-)} - a = —a, 
dn dn 
and so the leading terms in (6’4) and (6’5) arise from taking 
2a = TT—0, or n + ^ = 2 sin ^9. 
The corresponding value of the index o- is then 
o-Q = 2i/^ + n 7 r —w = 22 sin a+^TT— {2n + l) a + n-n-— (ri + -|-) 9 + ^7r, 
= 22 cos |-0 + -j7r. 
To determine the form of the index near to this special value of n we take 
d^a- _ • (yda _ 2 _ 2 ^ 
dn^ dn z sin a z cos ^9 
and then we find the approximate formulse 
(T = 0-0 + 
(6-51) < 
where 
2 COS 1-0 
Wo + |- = 2 sin |-0. 
