494 
MR. G. H. DARWIN ON A PLANET 
Now consider the single satellite m, c, SI, &c. 
If this satellite alone were to raise a tide in the planet, the planet would be dis¬ 
torted into an ellipsoid with three unequal axes, and in consequence of the postulated 
internal friction, the major axis of the equatorial section of the planet would be 
directed to a point somewhat in advance of the satellite in its orbit. 
Let f be the angle made by this major axis with the satellite’s radius vector ; f is 
then a symbol subject to suffixes 1, 2, 3, &c., because it will be different for each 
satellite of the system. 
Then it is proved in (22) of my paper on the “ Precession of a Viscous Spheroid,”'" 
2 
that the tidal frictional couple due to this satellite’s attraction is C sin 4f. 
Now it appears from Sec. 14 of the same paper that the tidal reaction, which affects 
the motion of each satellite, is independent of the tides raised by all the other satellites. 
Hence the principle of conservation of momentum enables us to state, that the rate 
of increase of the orbital momentum of any satellite is equal to the rate of the loss of 
rotational momentum of the planet which is caused by that satellite alone. Tire rate 
of loss of this latter momentum is of course equal to the above tidal frictional couple. 
When the planet is reduced to rest the orbital momentum of the satellite in the 
circular orbit is Sic 2 Mmj Hence the equation of tidal reaction, which gives 
the rate of change in the satellite’s distance, is 
d Mm Q 
— —- SI C 3 
dt\M -f m 
sin 4f 
A similar equation will hold true for each satellite of the system. 
This equation will now be transformed. 
By Kepler’s law Sl 2 c 3 = p (M-\-m) and therefore 
( 1 ) 
Mm 0 , Mm , 
—- Slc 2 =ix *—-—c* 
M+m r (M+ m) J 
By the theory of the tides of a viscous spheroid (Phil. Trans., Part I., 1879, p. 13) 
Hence 
Hence (1) becomes 
tan 2f=-~——, where 2p = 
* ac 2(ro-/2)/p , 3 _ 
&m 4t i + (n-siy~if’ also 7 ~~ 
2 gaio 
19i/ 
fi)Mm dr) , 3 >3 C (firn)~ (n — /2)/jp 
(M+m) h - dt '' 2 ' g c 6 l + (?i—/2) 2 /p 2 
( 2 ) 
Now let Ch be the angular momentum of the whole system, namely that due to the 
*' Phil. Trans., Part II., 1879, p. 459. 
