of the Evolution of the Earth-Moon System. 429 



expressed : — 



<\A) a ^r> (1) 



where 



r= distance between centres of earth and moon, 

 £=time elapsed from a fixed point, 

 p(n-n) 



i+f( n -ay> KZ) 



n = angular velocity of earth's rotation, 

 11 = angular velocity of moon's orbital revolution, 

 /> = quantity varying inversely as the viscosity of the planet. 



The extreme interest of equation ( 1) consists in the appear- 

 ance of the inverse sixth power of the distance. 



As the function M* varies very slowly, we find by integra- 

 tion, for any portion of time during which ^ may be regarded 

 as constant, 



t=Ar~*- + B, (3) 



a most unexpected and remarkable result. 



Upon reading Mr. Darwin's papers, my mind turned to a 

 problem with which I was familiar, viz. the retardation of the 

 earth's rotation produced by the lunisolar tide exerted upon 

 the ocean supposed collected in an equatorial canal, the moon 

 and sun having no declination; and I readily found an equa- 

 tion to express the evolution of the earth-moon system, on the 

 foregoing hypothesis as to friction. 



This equation is the following : — 



^)^?> (±) 



where 



ltl Vo(n-fl) (5 



f= coefficient of friction supposed proportional to relative 



velocity, 

 h varies inversely as r 3 , 

 Y = velocity at earth's equator. 



This leads, as in Mr. Darwin's hypothesis of viscous earth, 

 to the integral 



*=AvV+» (6) 



The form of the functions M* and <E> is similar, as both ascend 

 by odd powers of (n— 12) and vanish when n = Q — that is to 

 say, at the beginning and end of the evolution by friction of 

 the earth-mo om system. 



