118 THE TIDAL PROBLEM. 



The first equation can be satisfied only by i = (the case i = 7z makes only 

 a change in direction of the axes in the final solution). Then the second 

 one gives, since P can not equal infinity, as the conditions for a maximum 

 or a minimum 



i=0 P5-MP + mi=0 (117) 



Then the first equation of (116) shows that either ii = or D=oo . In the 

 latter case i^ is indeterminate. 



To investigate whether the roots of (117) correspond to a maximum 

 or minimum we form (at i = 0) 



The right member of the first equation is positive for both roots of (117). 

 We must consider the function 



For the smaller root of (117) the right member of this equation is negative, 

 and therefore E is neither a maximum nor a minimum for this set of values. 

 For the larger root of (117) it is positive and the corresponding value of E 

 is a true minimum. Whenever the system arrives so near the condition 

 corresponding to this point that the equi-energy curves are closed they 

 remain closed for all smaller values of E down to the minimum, and under 

 the influence of tidal friction the system will inevitably approach this 

 condition of minimum E, and having attained it will remain there. 



It follows from the second of (116) that for either value of P satisfying 

 (117) we have D = P. 



