496 
MR. G. H. DARWIN ON A PLANET 
Now let 
J- 
M 
\M+m 
(7) 
where a is any constant length, which it may be convenient to take either as equal 
to the mean radius of the planet, or as the distance of some one of the satellites 
at some fixed epoch. £ is different for each satellite and is subject to the suffixes 
1, 2, 3, &c. 
The equation (6) may be written 
I J AY 7 c 3 A‘ = _ ( Bf x49 x iAA[A±AD « 
\M+m) it ' 2 ' X ‘ X J/gi>\ M ) 7 c 3 6 (c>) 
Now let A 
N = (§) : x 49 X 
fiC 2 
ifAgp 
( 8 ) 
And we have 
bE 
dt~~ A blj 
(9) 
[In order to calculate A it may be convenient to develop its expression further. 
so that jg-| 
\a 
and 
^=(1) (1)49 
(«/«) 7 
where p = 
gaw 
I9u 
( 10 ) 
Since p is an angular velocity A is a period of time, and A is the same for all the 
satellites.] 
In (9) £is the variable, but it will be convenient to introduce an auxiliary variable x, 
such that 
or 
(ii) 
Then 
/AMm, i fAMMvuA 
(AT+m y C ’ = (ilf+mfi X 
Let 
/AM'iimA 
C(M+mf 
( 12 ) 
k is different for each satellite and is subject to suffixes 1, 2, 3, &c. 
Thus (3) may be written 
h—n-\r^KX .(13) 
. . iiMm uil/bft 1 
Again “— — 
o ( il /+ m)’a X- 
