27] OF THE MOON'S MEAN MOTION, ETC. 213 



Hence our equations become 

 d*u p 1 n H ( 3 H , 





- e' 2 cos (20 - 2n't) +-e' cos (20 - 3n't) 



-\t- cos (2(9 -n't)\ 



7 



- 6/ sin 2 ^~ 2 " + e/ sin 2 * - 3w/ ' 



and 



d 



~ ' sin ( 2 ^ ~ 2? ^) + <'' sin (20- 3n't) 

 u 



2. After these preliminaries, it will be convenient to begin by finding 

 the relations between the actual mean motion n of the Moon and the 

 constant parts of u and H" when these quantities are developed in the 

 form we have adopted, carrying the approximation as far as terms involving 

 m*e'~, on the supposition that e' and therefore also that n is constant. 



For this purpose it is sufficient to take 



nt + e = 0+ 3me' sin n't - m 2 ( 1 -| e' 2 ) sin (20 - 2n't) 



o \ 2 / 



- m-e' sin (20 - 3n't) + mV sin (20 - n't), 



U--41-- ?J^ 2 e' cos n't + m* (l - - e' 2 j cos (20 - 2n't) 



7 1 1 



+ - rri'e' cos (20 - 3n't) - - mV cos (20 - n't) \ , 



Zi ] 



which are readily derived from the equations of motion. 



