504 
MR. G. H. DARWIN ON A PLANET 
If the second or ^-satellite be larger than the first or cc-satellite the latter of these 
momenta is larger than the first. 
Now Ch is the whole angular momentum of the system, and in order that there 
may be maxima and minima determined by the equations bz/bx= 0, bz/by= 0, the 
equations (26) must have real roots. Then on putting y equal to zero in the first of 
the above conditions, and x equal to zero in the second we get the following results:— 
First, there are no maxima and minima points for sections of the energy surface 
either parallel to x or y, if the whole momentum of the system be less than 4/3* times 
the orbital momentum of the smaller or ^-satellite when moving at distance y 1 . 
Second, there are maxima and minima points for sections parallel to x, but not for 
sections parallel to y, if the whole momentum be greater than 4/3* times the orbital 
momentum of the smaller or x-satellite when moving at distance y Y , but less than 
4/3' times the orbital momentum of the larger or y-satellite when moving at 
distance y 3 . 
Third, there are maxima and minima for both sections, if the whole momentum 
be greater than 4/3* times the orbital momentum of the larger or y-satellite when 
moving at distance y 3 . 
This third case now requires further subdivision, according as whether there are not 
or are absolute maximum or minimum points on the surface. 
If there are such points the two equations (24) or (25) must be simultaneously 
satisfied. 
Hence we must have n = /2. l .=L2 y , in order that there may be a maximum or 
minimum point on the surface. 
But in this case the two satellites revolve in the same periodic time, and may be 
deemed to be rigidly connected together, and also rigidly connected with the planet. 
Hence the configurations of maximum or minimum energy are such that all three 
bodies move as though rigidly connected together. 
The simultaneous satisfaction of (24) necessitates that 
X 1 — 
A/^2 3 3 
or r- 
Xo K-, s 
\~x T 
/C.t Aj 
Hence the equations (24) become 
Ji¬ 
ll- 
These equations may be written 
h 
\Xj/c 2 
*1 + K-2 
+'<2 
1 
/q x r 
A, 1 
r= 
W- 
(r) 4 - 
K i + /^(Xg/Cipqtfjj)' 
h 
t(x ; ) 3 - 
« 3 y r 
+ K d~^'2 K l l\ K s) 
, = 0 
1 
(r) 3 - 
A)/« 0 
y 
= 0 
/c 1 (X ] a: 2 /X 2 « : 1 )“ + k 2 ^ ~« 1 (Xi/e 2 /X 2 «: 1 )' i + /c 2 
(27) 
