1880.] History of Planet and Single Satellite. 



257 



Again, since C = f Ifa 2 , and since C is unity, therefore M=|/a 2 , where 

 if is estimated in the mass unit. 



Therefore Mm/(M-t-m) is unity, when M and m are estimated in 

 the mass unit, with the proposed units of length, time, and mass. 



According to the theory of elliptic motion, the moment of momentum 

 of the orbital motion of the planet and satellite about their common 

 Mm , 



centre of inertia is — Qc- v 1 — e 2 . Now it has been shown that the 

 M+m 



factor involving If is unity, and by (1) Qc 2 =Q~i=c^. 



Hence, if we neglect the square of the eccentricity e, the m. of m. of 

 orbital motion is numerically equal to Q or c*. 



Let x=Q~*=c-. 



In this paper x, the moment of momentum of orbital motion, will 

 be taken as the independent variable. In interpreting the figures 

 given below it will be useful to remember that it is also equal to the 

 square root of the mean distance. 



The moment of momentum of the planet's rotation is equal to Qn ; 

 and since C is unity, n will be either the m. of m. of the planet's rotation, 

 or the angular velocity of rotation itself. 



With the proposed units r=f m/c 3 =-|a 2 a3~ 6 , since m=-|a 2 ; and 



Also T 2 /g (a quantity which occurs below) is equal to f a^/vx 12 . 



Now let t be the time, and let 2/ be the phase-retardation of the 

 tide which I have elsewhere called the sidereal semi-diurnal tide of 

 speed 2n, which tide is known in the British Association Report on 

 Tides as the faster of the two K tides. 



Then if the planet be a fluid of small viscosity, the following are 

 the differential equations which give the secular changes in the 

 elements of the system : 



rf "'=->. T3 s in.t/(l-. Q - N ) (2). 



> *^*/.K) < 3 >- 



1= i^/(;±9H?) • • • ■ * 



!=-^sin4/ii±i) (5) . 



at g x 



lde , T 2 . l/ n 18Q\ 



~jT = i— sm4/.-ni- ..... (6). 



e dt (J x \ n / 



The first three of these equations are in effect established in my 

 paper on the " Precession of a Viscous Spheroid,"* § 17, p. 497, eq. (80). 



* " Phil. Trans.," Part II, 1879. 



