1879.] On the Secular Effects of Tidal Friction. 169 



It will be supposed that the figure of the planet and the distribution 

 •of its internal density are such that the attraction of the satellite causes 

 no couple about any axis perpendicular to that of rotation. 



Then the two bodies revolve in circles about their common centre of 

 inertia with an angular velocity Q, and, therefore, the m. of m. of 

 orbital motion is 



\M+m/ \M+mJ M+m 



Let /u be attraction between unit masses at unit distance. 

 Then, by the law of periodic times, in a circular orbit, Qh^=fi(M+m) 

 whence 



And the m. of m. of orbital motion =/u,%Mm(M+m)~iQ~*. 

 The m. of m. of the planet's rotation is Cn, 



and therefore h=C | n+ fi^^-(M + m)~iQ~* j> . . . (1). 



Again, the kinetic energy of orbital motion is 



2 \M+mJ 2 \M+m) 2 M+m ^ v } 



The kinetic energy of the planet's rotation is \Cn 2 . 

 The potential energy of the system is 



r 



Adding the three energies N together 



2e=0 1 n^-^^(M+m)-mi j . . . (2). 



Now, suppose that by a proper choice of the unit of time, 

 is unity, and that by a proper choice of units of 



length or of mass C is unity,* and 



Let »=G~i, y—n, Y=2e. 



M 



* Let v— — , then if g be the mean gravity at the surface of the planet, and if a 



m 



be its mean radius, 



v 



and ^mm(M+m)^=^ffa^y^ = Ma^ { (1 + v) } 1 



Then if the planet be homogeneous, and differ infinitesimally from a sphere, 

 C=%Ma 2 , andj 



