94 THE TIDAL PROBLEM. 



and the potential energy of mj and m^ are respectively 



(rotation) ^ ^^ (revolution) ^j (potential) — 27r 



Then from equations (32), (33), and (34) we find 



(rot.) (rev.) (pot.) 



dEm, ^ P dE dEm,+^n, ^ D dE dEn.+m, ^ -2D dE (35) 

 dt P-D dt dt P—D dt dt P-D dt 



whence 



(rot.) / (rev.) (rot.) / (pot.) 



dEjn, I dE^m,+m,) _ P dE^ / dEm.+m, ^ P (36) 



dt / dt D dt I dt 2D 



When the directions of the revolution and of the rotation are the same, 

 the loss of energy of rotation is to that of revolution of both bodies as 

 the period of revolution is to that of rotation, and the potential energy 

 gains twice the loss of the revolutional energy. 



The number of periods of rotation in one of revolution is, from (23), 



iV=^=~(M-Pi) (37) 



D nil 



When (28) is satisfied N = l. The maximum value of N is defined by 



m,^ = (M-^Pl) = (38) 



whence, at maximum N, 



64 256 w, 



