114 THE TIDAL PROBLEM. 



somewhat like parabolas and vanish at — oo for EM'^=—z. They are the 

 analytic continuations of the curves on the right, the union being at infinity. 



When K recedes to — oc as jE'= — it reappears at +00 as jE^ continues 

 to decrease. 



It follows from (105) and (106) that ?/i + 2/2 < 1 ^or ^^^ negative values of E. 

 Consequently for all points on the closed ovals we have |el<l, and the 

 system can not be wrecked by a collision of the bodies. Therefore, if at 

 any time the configuration of the system corresponds to any point on one 

 of the closed ovals to the right of A , it will always tend with decreasing E 

 through tidal friction toward the configuration corresponding to the point B, 

 and the evolution will end with this configuration. Because of the sym- 

 metry of fig. 15 with respect to the line e = 0, it follows that if at any 

 time e = it will always remain zero. 



While under certain conditions the system will inevitably progress 

 toward a definite configuration, in the general case it is not possible, with- 

 out hypotheses as to the physical condition of the bodies, to determine the 

 character of the evolution. The question of greatest interest in the present 

 connection arises in the case where the conditions lead to doubtful results. 



If the moon separated from the earth by fission and if its orbit were 

 originally circular, it would not become elliptic through tidal friction. Since 

 the orbit is now considerably eccentric, we must assume that it was some- 

 what eccentric at the time of separation. Consequently, let us suppose 

 the moon has just separated from the earth so that P = D, and, supposing 

 that e^O, let us find whether the moon will fall again to the earth or recede 

 from it. Since the orbital velocity will have been such that the moon's 

 motion will have fulfilled the law of areas, while the rotational velocity 

 will have been uniform, there will have been relative motion of the various 

 parts of the system, and consequently tidal friction. We are to find the 

 effects of this loss of energy on the distance of the moon. 



Under what seem reasonable assumptions we have seen in section V, 

 equations (33), that when the orbits are circular the rate of change of the 

 month is given by 



dP^c P-D 

 dt ~r« P^D 



We shall now assume that when the orbit is elUptic the rate of loss of 

 energy at any instant depends upon the square of the product of the tide- 

 raising force and the angular velocity of the tide over the earth. This 

 assumption is equivalent to taking the circular case as applying instan- 

 taneously to the elliptic case, and omits the lag in tidal conditions due to 

 inertia. With this assumption the equation above becomes 



dP c' e-io ^ ,.... 



dt ~ r" di ^ ^ 



where c' is a positive constant. 



* This result agrees with that found by Darwin in 2, p. 497, eq. (79) after change of 

 variables, notation, and proper speciaUzation of his problem. 



