530 
PKOFE8SOK (1. H. DAFnVFX OX EIJ.FF-'KOIF)AF. HAR5F0XIC AXAL5"SIS. 
At present we only require this to the first power of and since jFjC?^ does not 
contain cos 2(f>, the expression (75) as far as at present needed is simply unity. 
Ao-ain, by (38) of § 9, 
therefore by multiplication 
(1—cos 20)- 
T1 lis is also unity to the order at present needed. 
Hence 
{1+3“2/5"(/ —6)} —l)cos2(^} . . . .(76). 
J P 
jc;)' [w.p-(p.»o= i", Mil-nfiifi [j+ 1 -(■2i+i)] 
A-/3-[5/- +14/+12 - 4 (2i+1) (y+3)]} 
47r]\I 
1 
4-6\-^"(5^' —2j —4) 
-7rM{2/3+i/3^(j-lH 
(77). 
§ 18. Preparation for the Integrations when ,s=l and 0. 
We have now to evaluate the three integrals 
o 
L=|p(S'P.‘)Oi<r, 
N = j'p(C,P,)-rf<r 
a+/3 
—sin® B) do-, \ . 
J 
(78). 
and from these to determine three others when S, C replace C. 
We have 
pd„ _ \I r 
ddd<j>~\ (l-^J J I (1-/9 cos 20)4 J’ 
d6 d(f) 
cos= 0 + ^'ZT^ -\ 
(1—/9 cos 20)- J 
It is the second factor which alone involves (f), and as I shall now first integrate 
with respect to (f), the first factor may be dropped for the moment, and the second 
factor multiplied by the squares of the cosine or sine functions. Since the integration 
is from = 2;? to 0, those terms wliich vanish on integration may be dropped. 
