THE SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 125 



This is 149 miles less than that found when the action of the sun's tides 

 upon the rotation of the earth was neglected. That is, when the effects 

 of the sun's tides upon the rotation of the earth are included, it is found 

 that the possible initial distance of the moon is not materially diminished, 

 and that the theory of the jSssion of the earth and moon is still subject to 

 the serious embarrassment of wide separation immediately after the sup- 

 posed division into distinct masses. 



XV. THE SECULAR ACCELERATION OF THE MOON'S 



MEAN MOTION. 



As is well known, there is a secular acceleration of the moon's mean 

 motion of about 4" per century which has not been explained by the ordi- 

 nary perturbation theory. It was long ago suggested by Delaunay that 

 it may be due to tidal friction, and Darwin has made an investigation of 

 the subject in 2, section 14. 



If we accept the tidal explanation, the apparent acceleration of 4" is 

 due to an actual retardation of the moon, the only result possible accord- 

 ing to (32), and a greater retardation of the rotation of the earth. Since 

 the rotation of the earth is used to measure time, the period of revolution 

 of the moon on this basis apparently is accelerated. In making the discua- 

 sion we shall neglect the effects of Oj, e, I'l, i^ and S. 



Let — A^i be the gain in longitude of the moon in a century, and —Av^ 

 the corresponding gain in the angular distance of rotation of a meridian 

 of the earth. Then we have 



Av2-Avi=4" 



dP _ 2r. dd _27c Av^ _P^ /\v^ 



dt d^ dt e^ (lOOPO' 2;r (lOOP')' 



dP _ 27C doj _27t AVg _D^ Ar^ 

 dt ~ io^ dt~ (x)^ (lOOP')' ~ 2^ (TOOFP 



From these equations and (32) we find 



dD PiD^ 4" dP 3TOiP2 4" 



(138) 



dt 27r(P5-3mJ (lOOP')' dt 27r(P5-3m,) (lOOP')' 



(139) 



Representing the value of P at < = fo by P^, integrating the second equa- 

 tion, and determining the constant of integration, we have 



The present rate of tidal evolution of the earth-moon system depends 

 upon the forces acting and upon the physical condition of these bodies. 

 If we regard the 4" per century of apparent gain in longitude of the moon 



JET 



as due to tidal evolution, we have a measure of the -3- of equations (32). 



