OP ARTS AND SCIENCES : FEBRUARY 14, 1865. 391 



rotatory velocity relative to the moon is diminished, in round numbers, 

 the s-^TTiTTjffTrTrir P^^^ "^ ^ century. Now if we suppose the moon to 

 have been originally in a fluid state, it may be demonstrated that the 

 effect of the earth in retarding the moon's rotatory velocity would have 

 been to that of the moon in retarding the earth's rotatory velocity, all 

 other conditions being the same, as the square of the earth's mass is to 

 that of the moon, or as 6,400 to unity, assuming the mass of the moon 

 to be ■g^ih of that of the earth. But the density of the tidal wave on 

 the earth is much less than the mean density of the earth, and also 

 than the density of the earth's surface, and it moreover occupies only 

 about three fourths of its surface, so that we may assume, if the moon 

 be considered homogeneous or nearly so, that the effect of the earth 

 upon the moon must have been, at least, 20,000 times greater than the 

 effect of the moon upon the earth in diminishing the rotatory velocity. 

 Hence, if we assume that the rotatory velocity of the moon originally 

 was equal to that of the earth, and its tidal wave was displaced two de- 

 grees on its surface by friction, it would have lost ij^jjij part of its 

 rotatory velocity relative to the earth in a century, and consequently, 

 if the amount of tidal displacement could have remained the same, the 

 time of the moon's rotation on its axis would have been reduced to the 

 time of its revolution in its orbit, as we now find it, in 14,000 centuries. 

 But if we suppose friction to diminish with the velocity of rotation rela- 

 tive to the earth, the amount of tidal displacement would diminish in 

 the same ratio, so that if we put v for the rotatory velocity of the moon 

 relative to the earth at any time t, the original velocity being consid- 

 ered unity, we get from the preceding condition, 



dt = —14,000-". 



V 



Putting m for the modulus of common logarithms, this gives 



rp 14,000 , 



T = j^ log. v', 



T being the time in which the original rotatory velocity would be re- 

 duced to some given velocity v'. 



From the preceding equation it is seen that the rotatory velocity of 

 the moon relative to the earth, or to the time of its revolution, could 

 never have been reduced to nothing, however lonof the moon might 

 have remained in a fluid state, but that it could have been reduced 

 within any assigned limits, however small. Now Laplace has shown, 

 that in order that the mean relative velocity should become nothing, 



