152 THE TIDAL PROBLEM. 



readily assignable which could have increased it. Among those which may 

 have decreased it apparently the solar tidal friction is the only one which 

 can have produced sensible results. 



The effect of the moon's tides in the earth has been to transfer moment 

 of momentum from the earth to the moon. According to the results given 

 in the preceding paper on tidal evolution, section XIV, the maximum 

 moment of momentum the earth-moon system could have had, including all 

 that the sun could have taken from it, is that belonging to the system mov- 

 ing as a rigid mass with a period of 4.8 hrs. Let P represent this common 

 period. Then, neglecting the inclinations of the planes of the equators of 

 the earth and moon to the plane of their orbit and the possible eccentricity 

 of their orbit, factors which reduce the moment of momentum, we have by 

 equation (6), loc. cit., 



{2tz) -^ {nil + ^2) ' " Jr 



The quantities c^ and Cj depend upon the distribution of mass through- 

 out Wi and m^, m^ representing the earth, and Wj the moon. If the masses 

 are homogeneous 0^ = 0^ = A. If they obey the Laplacian law of density 

 Cj = C2 = 0.336. If the distribution of mass is such that the densities increase 

 from the surfaces to the centers of these bodies, the values of c^ and c^ are 

 less than 0.4. We shall certainly get too large a value for M by putting 

 Ci = C2 = 0.4. Adopting these values we find that for the earth-moon system, 

 in the units of this paper, ilf = 8X10~". 



Since large moment of momentum tends to instability we shall favor the 

 theory of fission if we add 25 per cent to this number, supposing that per- 

 haps this amount may have been lost through meteoric or other friction. 

 Then, using ilf =10-^^ and m = 3 X 10~^ equation (6) gives for this united 

 earth-moon mass at the time when the Jacobian ellipsoid branched a mean 

 density 215 times that of water. We find similarly that this hypothetical 

 earth-moon mass could not become even so oblate as Saturn is now until 

 its density had become 10.4 times that of water. Since the present density 

 of the earth is only 5.53, this means that if the hypotheses upon which this 

 computation was made are valid, the earth-moon s)^stem can not have arisen 

 from the fission of a parent mass under the influence of rapid rotation. 



In the preceding paper, starting with the earth-moon system as it now 

 exists and following backward in time the effects of tidal friction, it was not 

 possible to get the earth and moon in close enough proximity to make the 

 fission theory seem possible. Now, starting with the supposed initial sys- 

 tem with the critical factor, the moment of momentum, determined from 

 observations, we do not find the figure approaching an unstable form until 

 the density is more than 40 times the present density of the earth. 



