P = = period of revolution of nii and Wj 



84 THE TIDAL PROBLEM. 



II. GENERAL EQUATIONS. 



The problem treated will be that of the tidal interactions of m^ and m^, 

 which, in certain cases, are supposed to be disturbed by a large distant 

 mass S. The masses Wj and m^ will be assumed to be spherical. Then let 



aj = radius of m^; 03 = radius of m^ 

 r = distance from the center of nii to the center of Wj 



-0, = ~ — r = distance from m, to center of gravity of m, and m, 



/), = ^ — r = distance from w, to center of gravity of m. and w, 



X, y, z = set of fixed axes with origin at center of gravity of m^ and m^ 

 t = inclination of orbits of m^ and m^ to a;t/-plane 

 i\ = inclination of plane of equator of m^ to a^y-plane 

 t2= inclination of plane of equator of m^ to a;y-plane 

 e = eccentricity of relative orbit of w, and Wj 

 a = major semi-axis of relative orbit of w^ and Wj 

 6 = angular velocity of revolution of m^ and m^ 



±27t 



av.d 



a>, = angular velocity of rotation of w^ 

 a>2 = angular velocity of rotation of m^ 



Dj = — — = period of rotation of m^ , 



-L.2i7t 



D2 = — = period of rotation of Wj 



a>2 



S = & distant disturbing mass 



r' = distance of S from center of gravity of lUi and mj 

 P' = period of revolution of S 

 M, M', M" = whole moment of momentum of system, exclusive of S, 



about the z, x, and y-axes respectively 

 ^ = whole energy, both kinetic and potential, of system exclu- 

 sive of ;S 



The moment of momentum of m^ about the 2-axis is the moment of 



momentum about its axis of rotation multiplied by cos i^ plus its moment 



of momentum of revolution around the z-axis. The rotational moment 



of momentum of Wj around an axis through its center and parallel to the 



2-axis is 



c-jn^aiOJi cos ij 



where c^ is a constant depending upon the law of density. The corre- 

 sponding quantity for Wj is 



02^112(120)2 cos 1*2 



The moment of momentum of revolution of Wj about the 2-axis is 



2 



w, p,^ 6 cos i = 7 — ^, ^ ,, r^ 6 cos i 

 {m^-\-m2) 



