336 Dr. T. H. Havelock on 
ultimately, but at present it is regarded as indefinitely larger 
than p. 
Then we have 
_2 C 2 cos 26 d6 2 f _cos 20 d6 
ttJ o vV + 4cos~20 + ~ J v> 2 + 4sii 
1 2 
= --^1+-^ (13) 
In 3/j we write, as far as terms in p 4 , 
2 cos 2(9 cos 20 
L 8 cos 2 ^ 128 cos 4 0j 
:j2 
V> 2 + 4cos 2 <9 costf 
Then we integrate the terms separately, substitute the 
7T 
imits and ~ — e, and assuming e small we expand as far as 
necessary; we obtain thus 
3 4/ 133 7 1 5, 1 \ /1ylN 
+ i28^("lii0 + l2?-4?-8 lo «2 e > * ' (U) 
In the second integral y 2 , p is small compared with 6 
throughout the range ; we substitute for cos 26 and sin 6 
their expansions in powers of 6 and expand by the binomial 
theorem. We obtain 
-fn-^- 2 fl* ■ 2 ** 64 ^ 2 fl 8 -[ ^ 
Jo L i "3 (r+ 3p 2 + 4^ 2 45p 2 +40 2 + 3(> 2 + 40 2 ) 2 J •jT+BP' 
including all parts which will give terms containing p* on 
integration. Integrating the parts separately and expanding 
as far as p^/e* we find 
1/ 1 3 . 15 A A 4e ^ p 2 3p 4 \ € 4 / 3^ 2 15;? 4 \ 
