204 
SIR G. H. DARWIN ON THE 
11 . Solution of the Equations. 
In all the ellipsoids of which we shall have to find the axes, it happens that K /2 tan 2 y 
is fairly small compared with unity. Hence it is possible to expand A in powers of 
that quantity. 
We have 
A 2 = 1 —k 2 sin 2 y = cos 2 y (1 + k 2 tan 2 y), 
and 
1_ _1_ 
A COS y 
[ 1 -J k /2 tan 2 y + K n tan 4 y 
A :j 
cos y 
[1 
■V 2 
tan 2 y - 
3 ' 5 K l 
2-4 
tan 4 y—...]. 
From this we easily obtain 
= I (! + 0 J = cot3 7 [y ( 3 + tan2 y) - 3 tan y]- 
fj I 
This is the well-known formula for the angular velocity of Maclatjrin’s ellipsoid. 
It should he remarked that (35) is identically satisfied by A = oo, k = 0, for when we use the above 
Values of e and </, the equation becomes divisible by 1 + t- 
Since £ is a symmetrical function of a, b, c and A, B, C, it follows that (is the same in form for A and 
for 1 /A. Therefore when we consider the case of A = 0, the formula (a) gives the required result, but 
c and y refer to the large body which is throughout most of this paper indicated by capital letters. 
Thus for A = 0, 
., a 2 sin 2 F n a 4 / sin 2 T \ 2 - a u / sin 2 T \ 3 
1(1 A cos 2/3 r + ** F fo^r) ^r^\co^Tj 
In the case of A = 0, k vanishes; ^ also vanishes and so also does the angle B. Hence we have 
6 = ( COS 2 /?, 7 ] = (. 
With these values, equation (35) becomes 
- Ad) [cos 2 y + (A + C) cos 2 p] = (3d - Ad) [3 + (A + 0 + cos 2 y]. 
Hence ( plays the part of an augmentation to A. 
With A = 0 the equation assumes the form 
(% - Ad) (cos 2 y + C cos 2 P) = (3d - Ad) (3 + C+ cos 2 y) . (b). 
It follows therefore that an infinitesimal satellite revolving about an oblate planet, whose rotation is the 
same as the revolution of the satellite, is very nearly identical in form with a small but finite satellite 
whose mass is a fraction of a spherical planet expressed by (. This curious conclusion follows from the fact 
that if we take equation (35) and put £ and >/ zero (which corresponds to a spherical planet and small 
satellite), we get exactly the equation ( b) just found, only with A in place of £. 
