120 THE TIDAL PROBLEM. 



XIV. CASE i = t\ = e = a2 = Sr^O. 



The sun affects the evolution of the earth-moon system in two ways. 

 First, its direct perturbing influence on the moon makes the month longer 

 than it would be if the moon were revolving at its present mean distance 

 in an undisturbed orbit. In the second place the tides which the sun 

 raises in the earth retard its rotation and reduce the moment of momentum 

 of the earth-moon system. We shall consider these two influences separately. 



The relation among the sum of the masses of the earth and moon, the 

 mean distance from the earth to the moon, and the moon's period is found 

 in the theory of the moon's motion to be 



p2 



a^[l+^(^)'. ■ . ]=k^m, + m,) (123) 



where P' is the length of the year. With this relation instead of having 

 equations (9) and (10) become, when e=i = i\ = a2 = 0, 



The disturbing forces which have made these changes in the equations 

 are mostly radial, and so far as the radial components are concerned can not 

 change the moment of momentum. The tangential components are periodic 

 with equal and symmetrical positive and negative values in a period. 

 Consequently the M will not have changed under these influences. 



At the time of the supposed separation of the earth and moon P = D, 

 and the first of (124) gives for the determination of P 



We find from the first of (124) that the value of M is in this case M = 

 3.63428. The smaller root of (125) for this value of M is Po = 0.205760, 

 and this period corresponds to an initial distance of i^o^ 9,206.2 miles, a 

 little greater than that found when the sun's action was neglected. 



Consider the direct tidal friction of the sun upon the earth-moon system. 

 The sun's tides lengthen D without producing a corresponding change in 

 the motion of the moon. Consequently in this case M is not constant. 



Let us assume, as before, that the amount of tidal friction is directly 

 proportional to the product of the square of the tide-raising force and the 

 square of the velocity of the tidal wave along the earth. It also depends 

 upon the physical condition of the earth. Then we have, including the 

 tides produced by both the moon and the sun. 





