25C> 



Mr. G. H. Darwin. 



[Mar. 18, 



tion and the eccentricity of satellite's orbit to the plane of reference are 

 treated as being small, and, lastly, it is supposed that the planet is 

 only attended by a single satellite. 



The satellite itself is treated as an attractive particle, and the planet 

 is supposed to be homogeneous. 



The notation adopted is made to agree as far as possible with that of 

 a previous paper, in which the subject was treated from a similarly 

 general point of view, but where it v\ as supposed that the equator and 

 orbit were co-planar, and the orbit necessarily circular.* 



The motion of the system is referred to the invariable plane, that is, 

 to the plane of maximum moment of momentum. 



The following is the notation adopted : — • 



For the planet : — 



M = mass ; a — mean radius ; g = mean pure gravity ; C = moment 

 of inertia (neglecting ellipticity of figure) ; n = angular velocity of 

 rotation ; i — obliquity of equator to invariable plane, considered as 

 small ; (J = -f^/a. 



For the satellite :— 



m — mass ; c — mean distance ; Q = mean motion ; e = eccentricity 

 of orbit, considered as small ; j = inclination of orbit, considered as 

 small ; t — -fra/c 3 , where m is measured in the astronomical unit. 



For both together: — 



v = M/m, the ratio of the masses ; s = t[(^^) 2 (1 + I ^ = the 

 resultant moment of momentum of the whole system ; E = the whole 

 energy, both kinetic and potential, of the system. 



By a proper choice of the units of length, mass, and time, the nota- 

 tion may be considerably simplified. 



Let the unit of length be such that Jf+m, when measured in the 

 astronomical unit, may be equal to unity. 



Let the unit of time be such that s or f[«' / /g , ) 3 (l + ' / )]3 may be 

 unity. 



Let the unit of mass be such that C, the planet's moment of 

 inertia, may be unity. 

 Then we have 



Q*c*=M+m=l (1). 



Now, if we put for g its value M/a 2 , and for v its value M/in, we 

 have 



s — -< . — — > = , since M + m is unity, 



5 I L m Mj m J 5 m 



and since s is unity, m = fct 2 , when m is estimated in the astronomical 

 unit. 



* " Determination of the Secular Effects of Tidal Friction by a Graphical Me- 

 thod," " Proc. Roy. Soc," No. 197, 1879. 



