APPLICATION OF SECTION V TO THE EARTH-MOON SYSTEM. 97 



a given volume will have been greater than that which we have used, and 

 we may adjust our formulas for it by increasing the c^ which occurs in the 

 original equations (6) and (7). It follows from (8) that the unit of length is 

 now greater than before. Then by (42) the numerical value of w^ is greater 

 than with the original Cj, from which it follows by (29) and fig. 11 that the 

 distance of the moon from the earth when the month and day were equal 

 was greater than that computed above. This would necessitate an increase 

 in the oblateness in order that the earth's equator should have extended 

 out to the moon, and the difficulty of having an earth already improbably 

 oblate is increased. 



Another hypothesis is that the earth was initially larger and, since the 

 separation of the moon, has shrunk to its present dimension. This is 

 quite in accord with the general ideas prevailing in the fission theory. We 

 can not apply directly the formulas which have been written down because 

 a change in volume would change the distance at which the system moved 

 as a rigid body. Consider an instantaneous change of any extent in the 

 radius of the earth. This does not change its rotational moment of mo- 



mentum. Then if we employ the same units jj is not changed; that is, D 



is changed so that when the new coefficient of m^ as defined in (8) and (42) 

 is used the quotient is constant. But an increase in the size of the earth 

 would result in an increase in D. Therefore wij is increased to fcrn, and the 

 condition for equality of the day and month is 



From this equation we find 



dP _ m^ 

 dk ~M-4PJ 



which is positive for the smaller root of PK Hence, if the earth has shrunk 

 from larger dimensions, the earth and moon moved as a rigid system at 

 a greater initial distance than that found above. That is, the hypothesis 

 that the earth has shrunk only adds to the embarrassment because of the 

 initial great distance of the moon. 



We may try the hypothesis that the moon separated from the earth at 

 a distance of 9,194.4 miles, that the earth's law of density was such that 

 at that time its radius was equal to this number, and that the moment 

 of momentum of the earth's rotation was the same as if its density were 

 as it is at present. The latter condition is necessary, for the whole moment 

 of momentum is unchanged by contraction. This hypothesis amounts to 

 simply attempting to change the law of density as well as the volume so 

 that the implications of the hypothesis shall be reasonably satisfied. 



We shall suppose the density is expressible by the Laplacian law, only 

 with different values of G and /^ from those which are used above. Letting 

 /2i = 9,194.4, the moment of momentum of the whole system was 



)i^ o // , N-^i' 2nc'mJ^ m2\ /9,194.4V 2 /Km 



M =2.c' (m,-f-m,) ^^^ =-^ (l +;-^^) [^^ a,^ (50) 



