478 MR. a. H. DARWIN ON THE PRECESSION OP A VISCOUS SPHEROID, 
But by (17) and (18) 
%' 
| 'Sill 71 —(- (G m s -j - COS 'll 
^ i — (B^ - b .tF w?OT/ ) COS 71— {~(G OT 2~f"'gGr WOT/ ) Sill II. 
Hence 
%' , M' • n l in dif 
- ( , cos rc+— sm « = G OT 2 +iG OTW/ = — »—. 
Thus 
Tr=C — cos 7+n sin .(49) 
In order to apply the ordinary formula for the motion of the moon, the earth must 
be reduced to rest, and therefore T must be augmented by the factor (M-\- in) -p M. 
Then if d be the moon’s longitude, the equation of motion of the moon is 
M + m 
M 
Tr 
(50) 
dS- 
But since the orbit is approximately circular —= 12. 
4 i i 9 o i -47 -P m 1~P r 
Also m = G -P iva z , and —7—- = . 
J 11 V 
Therefore by (49) and (50) 
0 ol + iHJS • , ■ -d l 
~ — = y vo- < — cos 1 -P n sm 1 — 
dt J v [ C 1 dt 
Now let £=(^~\ , whence /2 2 —/2 0 2 -p£ 6 . 
The suffix 0 to 12 indicates the value of 12 when the time is zero, and no confusion 
will arise by this second use of the symbol iq. 
But since the centrifugal force is equal to the attraction betwmen the two bodies, 
and the orbit is circular, therefore 12 2 r°=M-\-m. 
So that 12 0 h s =(M-]-m)£ 6 . 
Therefore 
r ~=(M J riny^f2 l ~ i , and 11 r = (M-\- 
and hence 1 
But M-\-m =because M and m are here measured in astronomical units of 
mass. 
