CASE X. 



115 



By hypothesis the mean value of 6 is equal to w, and we have therefore 

 from the two-body problem 



r = a[l — eco^(ji)t-\-- (1 — cos2w<) + 



•] 



(112) 



Consequently equation (111) becomes 



dP 14 43 



-— = — c"e [2 cos wf + -=- e + -^ e cos 2ojt + 

 at 3 6 



•] 



(113) 



where c" is a positive constant. The first and third terms produce no 

 secular results. The second shows that P secularly decreases. That is, 

 under the hypothesis that the loss of energy is proportional to the square of the 

 product of the velocity of the tidal wave and the magnitude of the tide-raising 

 force, it follows that if the moon had separated from the earth and originally 

 had been moving around it in a slightly eccentric orbit in a period equal to 

 that of the rotation of the earth, then the friction of the tides generated by the 

 moon in the earth would have brought the moon back to the earih.^ Since these 

 hypotheses certainly approximate the truth, we are led to the very probable 

 conclusion that the moon can not have separated from the earth in an elliptic 

 orbit and have been driven out to its present position by tidal friction. 



Precisely similar reasoning applies to the hypothesis of the fission of a 

 star into a pair of equal stars, and is an additional strong argument against 

 the soundness of this theory regarding the origin of binary stars, which 

 generally have large eccentricities. 



' Under the assumption that the planet is viscous and with different approximations, 

 Darwin's equations led to the same result. See 3, p. 854, eq. (292), also 3, p. 878 and p. 891 



8 



