200 
SIR G. H. DARWIN ON THE 
Since we may write — ^kxp in the form — (Ud — Ad) — ('Hi — Ad), the first of 
(30) in combination with (31) gives 
Hi-Ad = 
k 
3Ar 3 L 
<V 
l + 2c 2 + c 2 (1+ A) (1 + £) — jc 2 (2r+cr)] . . . (32). 
Referring to the values of t and <x in (27), I find 
-i (r-o-) = * §4’ [2- 3 (1 + k 2 ) sin 2 y] + * [4 (1 + k 2 ) - (5 + 6 K 2 + 5**) sin 2 y] 
IK / K 
+ to ^ [ 2K ' S cos2 y- * /2 sin2 y( 1+3R2 )] 
+ [4K /2 (1 + K 2 ) cos 2 y- k ' 2 sin 2 y (1 + 2K 2 + 5K 4 )] 
v K 
+ 
28 rVK 2 
[4 (1—k 2 K 2 ) cos 2 y —k ' 2 sin 2 y (1 + k 2 + K 2 +5k 2 K 2 )]. 
(2t + o-) = 1-4 4-2 L7 + 5k 2 - (3 + k 2 ) sin 2 y] + -^[11 + 6 k 2 + 7k 4 - (5 + 2k 2 + k 4 ) sin 2 y] 
jfe* 
7 ,2 K 
r k 
+ To -S [7 + 5K 3 - (3 + K ! ) sin 2 y] 
7’ K 
+ *-|!-,[ll+6K 2 +7K*-(5 + 2K 2 +K‘) sin 2 y ] 
Y iv 
+ 9 4¥4 [ 11 + 3« 2 +3K 2 + 7/c 2 K 2 -(o + K 2 + K 2 + /c 2 K 2 )sin 2 y] . . . (33). 
r kK 
In all the cases which we shall have to consider the first of these expressions is 
small compared with the second, because k is nearly equal to unity and k small, and 
because cos 2 y is also rather small. 
Now let 
e = — ^ (t— cr) + £ (1 + \) cos 2 /3 
rj = — 3 (2t + <t) + £ (l + A) 
. (33 bis). 
Then, since a — c cos 
y , b — c cos f3, the equations (31) and (32) become 
Hd - A d = (cos 2 y + A COS 2 (3 + e) 
J x > • • • • 
Hi - Ax 1 = 3 (3 + A+ cos 2 y + ??) 
OAT J 
(34). 
Eliminating kc 2 /S\r 3 , we have 
(^[j —Af) (cos 2 y+A cos 2 /S + e) = (^[i 1 —A/) (3 + A+ cos 2 y + ?7) . . (35). 
