130 THE TIDAL PROBLEM. 



moon could not have been less than 9,214 miles, a result more unfavorable 

 to the fission theory than that obtained when the eccentricity of the moon's 

 orbit was neglected. 



In sections XII and XIII it was assumed that the rotational moment 

 of momentum and energy of the moon and eccentricity of the moon's 

 orbit may be neglected, but the incHnation of the plane of the earth's 

 equator to the plane of the moon's orbit was taken into account. Under 

 these hypotheses it was found that the initial distance of the moon could 

 not have been less than 9,364 miles, a result more unfavorable to the fission 

 theory than any of those heretofore derived. 



All of the factors initially neglected and later taken up one by one 

 have made the initial distance greater than the originally computed 9,194 

 miles. Obviously all of them combined would operate in the same direction. 

 Since they only increase a difficulty which was in the first place serious, it 

 is not necessary to go to numerical results for all of them combined. 



The factors which remain to be considered in attempting to test the 

 fission theory by computing the initial distance of the moon are^ the sun's 

 perturbations of the moon's orbit and its effect upon the rotation of the 

 earth. The first part of section XIII is devoted to a discussion of the direct 

 action of the sun upon the moon's orbit, and it is shown there that includ- 

 ing this influence alone the initial distance of the moon could not have been 

 less than 9,206 miles, which is somewhat greater than that found when the 

 sun's action was neglected. 



The second part of section XIII treats the relative retardative effects 

 of the sun upon the rotation of the earth. The magnitude of the tide- 

 raising force of the sun compared to that of the moon can easily be com- 

 puted. Other things being equal there are good grounds for assuming that 

 the rate of tidal evolution is proportional to the square of the tide-raising 

 force. The friction depends also upon the speeds of the tidal waves with 

 respect to the earth's surface. At the present time the speeds of the moon's 

 and the sun's tides are about equal, but if we trace the system back until 

 the month and day were approximately equal this relation is no longer 

 approximately verified. We must resort, therefore, to some specific assump- 

 tion as to the way in which tidal friction depends upon the speeds of the 

 tides over the surface of the earth. The assumption was made that friction 

 is proportional to the first power of the velocity, and therefore that the 

 loss of energy is proportional to the second power of the velocity. It is 

 practically certain that this assumption will give results which are sensibly 

 true. Using this hypothesis and supposing that the rotational moment 

 of momentum and energy of the moon, the eccentricity of the moon's 

 orbit, the inclination of the plane of the earth's equator to the plane of 

 the moon's orbit, and the direct action of the sun on the moon's orbit may 

 all be neglected, it was found that the initial distance of the moon was 

 reduced from 9,194 miles to 9,045 miles. Thus it is seen that the one 

 factor which makes the moon's initial distance less than that found in the 

 first computation is not only of no particular consequence, but also that 

 it is less than some of the factors which increase it. Using all those factors 

 whose effects have been computed when they have been supposed to act 



