892 PROCEEDINGS OF THE AMERICAN ACADEMY 



or in other words, that the time of rotation should be equal to the time 

 of revolution in the case of the solid spheroid, it is not necessary that 

 these motions should have been exactly so at the origin of these mo- 

 tions, which is infinitely improbable, but only that tliey should have 

 differed by a very small quantity ; hence, however much these motions 

 may have differed originally, it was only necessary that the difference 

 should be reduced within certain very small limits by tidal action be- 

 fore solidification, in order that the mean motions should be made to 

 coincide afterwards. The reason of this is readily seen if we consider 

 that at the time of solidification, the longer axis of the spheroid would 

 be almost exactly directed toward the earth, and that at that time, if 

 there was still a very small velocity of rotation relative to the earth 

 remaining, the attraction of the earth upon the excess of matter of the 

 spheroid above that of the inscribed sphere, would bring the vertex 

 back before it would move through a quadrant, and thus prevent a ro- 

 tation and cause a libratory motion, the period of which, according to 

 Laplace, upon the hypothesis of homogeneity, would be about seven 

 years. But the motion would not only have to be too small to carry 

 the vertex of the spheroid through a quadrant, but too small to carry 

 it so far as to make the libratory motion observable from the earth ; 

 for, I believe, no such motion has been observed. If, with Laplace, we 

 assume that the coefficient of the term expressing the libration must be 

 6,000 centesimal seconds, or more, in order to be observable from the 

 earth, we can readily determine upon this hypothesis that the daily ro- 

 tatory velocity of the moon, at the time of solidification, must have been 

 reduced, at least, to uji^jVuzr part of a revolution. Substituting this 

 for v' in the preceding equation, we get 



_ 14^ 300,000. 



This equation gives, in round numbers, T' equal to about 175,000 cen- 

 turies. 



We know nothing with regard to the amount of rotatory motion 

 which the moon may have had originally, or with regai'd to the amount 

 of displacement of the vertex of the spheroid or tidal wave caused by 

 friction, and the preceding hypotheses upon which our results are 

 based have been assumed, not because they are considered the most 

 probable, but merely as a basis for results, from which approximate 

 results can readily be obtained for any other hypotheses. 



It is extremely improbable that the two motions should have origi- 



