206 
SIR G. II. DARWIN ON THE 
It would not be difficult to find the general expressions for these functions, but it 
does not seem worth while to do so. 
The equation (35) for determining the form of the ellipsoid involves the factor 
cos 2 y + A cos 2 /3 + e ; if we write 
M - A. sin 2 y 
(1 + A) COS 2 y + e ’ 
this factor may be written in the form [(1 + A) cos 2 y + e] [1 +Mk' 2 ]. Hence the 
equation (35) may be written 
[T 0 -T lK r2 +T 2 K n -7 3 K'\..] [(l + A) cos 2 y + e] [l+W 2 ] 
If now we put 
= «' 2 [o'o-o' 1 «' 2 +o- 2 /c' 4 -o- 3 /c' h ...] [3 + A + cos 2 y + 77]. 
"o = ^, i^-D, v 2 = —-- 2 -— 1 Ul5 U3 = T3_L3_.^ 
. 0-0 ^ T u °0 CT 0 <x 0 T 0 CT 0 cr 0 
we have 
v 2 - — v u 
r n —r 1 /c /2 + r 2 /c' i ... 
/2 , M 
cr 0 — o-j/c + cr 2 /< .. 
/4 
= v 0 [ 1 + Vi// 2 — V 2 K n + v 3 k' 6 
Hence our equation may be written 
° 3 + A + cos 2 y + ^^ + ^ ~ V2K ' 1 + V * K '*’‘ * ■H :1 + Mk ' 2 "> = K ' 2 - 
Whence on writing 
_ 3 + A + COS 2 y + rj 
(1 + A) cos 2 y + e 
k '2 _ 1 + ( Mv 1 — V 2 ) K ' — (Mv 2 — V :i ) k' 6 ... 
L/v 0 —M—v 1 
Ihe determination of L for given value of y involves that of rj and e, and these can 
only be found from an approximate preliminary solution of the whole problem. But 
when L is known approximately, the solution of (39) is very simple, for we first 
neglect the terms in k' 1 and k 1 on the right-hand side, and so determine a first 
approximation to k. As a fact I have not included the term in k' 6 in my computa¬ 
tions, because it would not make so much as l r difference in the value of cos -1 //. 
For IvOCHE s problem when e and rj are neglected the solution is very short, but 
when these terms are included the computation is laborious. 
We now turn to the determination of the radius vector. 
We have 
Since c 2 
F 
sin 2 /3 
and 
&1-A\ = 
k 3 COS y cos /3 _ 
sin 3 /3 
kc 2 
3X? 
Aa 3 
1+A 
(3 + A + cos 2 y + ? 7 ). 
, we have 
kc 2 _ 1 k sin y a 3 
3Ar 3 3 (1+A) cos (3 cos y r s 
