90 



THE TIDAL PROBLEM. 



The values of P and D must also satisfy equation (24). Starting from 

 any epoch, P and D must change, if at all, so that E shall decrease, for if 

 there is (tidal) friction in the system, some of its energy will degenerate 

 into heat and be dissipated. 



The relations between P and E and D and E are found from (23) and 

 (24) to be respectively 



(26) 



P?-2MP^ 



m, 



^1 -^ ■\r'> 



D^ 



E 



Z)2 (MD-m,y 7z 



(27) 



In these equations P may vary only from to + oo while D may vary 

 from — 00 to + 00 . 



The curve whose equation is (26) has two forms according as E, con- 

 sidered as a function of P, has a finite maximum and minimum or not. 

 In case there are a maximum and a minimum it has the form 7, fig. 10. 

 Since E can only decrease, it follows that if att=tg the period is on the part 



Fig. 10. 



of the curve ah it will decrease to the abscissa of the point b; if it is at b 

 it will permanently remain at that value; if it is between b and c it will 

 increase toward the value at 6; if it is at c it will remain there unless the 

 system suffers some exterior disturbance, when it will increase toward b 

 or decrease toward d according to the nature of the disturbance; if it is 

 between c and d it will continually decrease toward zero. 



When the curve has no finite maximum and minimum it has the form 

 77, and then whatever may be the value of P &t t^t^ it will continually 

 decrease toward zero with decreasing E. 



It is unnecessary to draw the curve whose equation is (27), for the 

 relation between the change in P and the change in D is given in (25). 



