196 
DE. T. J. FA. BEOMWICH ON THE 
The leading parts of M, N are accordingly given by the approximations 
( 6 - 6 ) 
where 
M = + cos 0-Q, N = — sin -Q, 
KV 
kv 
Q = 2 
y/( 2ri7r sin 0) 
and <7 has the value given in (6'51). 
The value of Q is approximately equal to the integral 
(6-61) 
where 
Loo y^(2?'io7r sin 0)’ 
n = Hq + x, o- = (Tq +x^I{z cos^O). 
Thus, approximately 
Q = 
9^0*0 
>;/(2^077 sin 6 ) 
V 
ttZ cos 40 
le 
I (22 cos P) 
2 sin 40 2 sin ^0 
Accordingly, to the same degree of approximatioD, we can take 
dO, 
(6’62) 
''t _ X„j^ 2 iz cos iff 
Then the components of force are given, as in (4’9), by 
Y= +Cy = 
aM 
1 aN 
(6-63) 
O • 7 ^ = COS (h - 7 -^; 
a0 Sin 0 d(p ^ kv do 
Z = —c/3 = — 
1 aM , aN 
sm 0 a^ do 
+ — = -sm<p 
dQ 
kv do 
where differential coefficients with respect to 0 are small compared with those with 
respect to 0, and so have been rejected. Thus, using (6'62), we have the approxi¬ 
mations to the forces in the scattered waves 
(67) 
Y= +cy 
cos 0 
2r 
^ 2iKacosifi—iKr 
Z = —cB = — sin 0 — c' 
^ ^ 2 r 
assuming that 0 is neither near to 0 nor to tt. 
When 0 is small, the approximation to P„ (cos 0) must be taken as 
and so 
P„ (cos 0 ) = Jo {(2n-|- 1 ) sin 40 }, 
sin 0P'„ (cos 0 ) = {n + ^) cos 40 Ji {(2n-l-1) sin 40}- 
