108 THE TIDAL PROBLEM. 



shape, we have proved rigorously that after fission, tidal friction can not 

 have increased their initial distance more than 200,000 miles. Such an 

 inconsiderable increase as this in the distance between two stars can have 

 had no important effects on binary systems. If the system started at the 

 configuration corresponding to maximum energy of the curve of fig. 10, 

 either it must have gone toward the condition of minimum energy just 

 computed, or the two bodies must have fallen together. 



Let us suppose the bodies to have shrunk as a consequence of loss of 

 heat so that their period of rotation would have become, except for tidal 

 friction, kD. Since the rotational moment of momentum is not changed 

 by shrinking, the wij becomes Km^, the mass changing its numerical value 

 because the definition of units depends upon the dimensions of the bodies. 

 Then equations (79) become 



^-^^ kD 71 ~ P^^ kW' ^^^^ 



Eliminating D we have 



E 1 , (M-Piy 



(90) 



71 pi 2/cmi 



The condition for a maximum or a minimum of E is 



P4-iWP + 2/cmi=0 (91) 



QP 2m 



From the derivative -t— = ^-?^i — h- it follows that, for given values of M 



OK iP^—M 



and Wj, the smaller k is the farther apart are the two roots of P. Let the 



common initial value of P and D be Po- Then equation (91) becomes, by 



(82) and (83), 



PJ - (1 +Ci)PoiP + Acc,Po» =0 (92) 



Since this equation is homogeneous in P and P^, its solution is of the form 



P§=/(c„/c)Poi (93) 



For a given amount of shrinkage of a body the constant k is determined, 

 because the moment of momentum is not changed by a decrease or increase 

 of volume. Hence, if we assume an initial Pq and the k, we may determine 

 the final (and greatest) P from (92), and compare the results with the data 

 given by observations of double stars. Or more simply, we may assume a 

 final P in accord with the data furnished by observations and compute the 

 K from 



P[(l+C,)Po4-Pi] 



K — 



c.P.^ 



(94) 



for various assumed values of Pq. Since the two stars are supposed to have 

 been initially in contact, the initial density may be expressed in terms of 

 Po, and the final density is determined by the initial density and the amount 



