214 
SIR G. H. DARWIN ON THE 
When s = 2, 
q 2 = l-^ + f k' 4 + tV 6 --- 
If we substitute these values for q 2 in (49), and express the functions of k 2 therein 
in terms of k' 2 , we obtain the following results :— 
0) = 7-Srd 1 + ( 1+ i"' 2 ++A K '“ • • •) 5 + W ( 1 + K ' a + + K ' 6 - • •) • • • 
7 KT 
¥ 
&*(-) = £-A i+t(i+i-c' 2 +M^+^ ,6 -)5 +lt(i+«' 2 +H^+fi'<' 6 -)p - [ ■ 
7#cV 
Shorter forms may be given to these by making the expansions run in powers of 
1 /kt 2 ; we then have 
®s (?)=W- 4 j 1 k ' 
,rj 7k 1 
K) 
“ 3^3 3 2 n ,4 
•• K r 
©X~) = rab+l(l + 0^ + 3V‘ + aV 6 - )5i + «( 1 + 0 *" + ***" + H«*-)^ 
\rj 7 ¥r 
Ki¬ 
lt will however suffice for our purposes to take 
(50). 
\r 
@3*(-) 
\r 
¥ 
7k 2 ¥ 
¥ 
7/cV 
1 
1 1 (1 O 
¥ 1 
/cr I 
1 + 4_A 
3 k¥ 
> 
(51). 
The next task is to determine the product ^ 3 S (r 0 ) W (1) C3 S (?7r) (5 = 0, 2). 
The form of the ellipsoid is determined by v 0 , where v 0 = — l — = . - _ . 
1 k sm y sm p 
If we write A x 2 = 1— q 2 &m 2 y, with the definition of p 3 s given above in (48), we 
have 
K (-.) = -£h 
k Sin' y 
It will be remembered that q has a different value according as s = 0 or 2. 
I now make the following definitions, 
W (p) = p (k 2 p 2 - 2 2 ), C 3 s ((f)) = ( q' 2 -K 2 cos 2 <f>) v/(l ~k' 2 cos 2 <j>); 
(52). 
so that 
W(l)C,*(M = «'V-2’).03). 
It is eas}^ to show that rigorously 
q' 2 (K 2 -q 2 ) = l s [l- K ' 2 - K ri ±(l-^ 2 ) x /(l-K f2 +^)] (s =0, 2). 
Whence approximately, with the upper sign for 5 = 0, 
q"(¥-q 2 ) = A(l-*' 2 +iV 4 +0*' 6 ...) 
(54). 
