27] OF THE MOON'S MEAN MOTION, ETC. 219 



d*a /day de' 



neglecting ^ , (-%) and also m ^ 



in the coefficients of the periodic terms. 



d*u 

 Hence -r + u 



contains the additional terms 



1 ( dp' de' de' 1 



+ - i- \ 20mV - sin (26 - 2n't) - 1 4m 2 , - sin (20- 3n't) + 2m' -.- sin (20- n't) [ . 

 an\ dt at at '} 



Also -Jt^ contains the additional term -- 4 [ 2817]. 



_i 7i Ct 



The other terms which enter into the first differential equation receive 

 no additional terms of the order to which we restrict ourselves. 



6. Also differentiating the expression for H*, and including terms of 



de' 

 the order mV-^- in the non-periodic part, but only those of the orders 



,de' , . de' 



m 2 -j- and mV -j-- 



dt dt 



in the periodic part, we have the following additional terms in \ a , viz. 



(it/ 



1 (2dn Ida 2421 4 ,de' 15 n e'de' ., 



na TT~> \ J* + -^TT + ~^T me ;7 - !T m " ^7" cos ( 2<? ~ 2n 

 a adt 32 a< 2 rf< 



~ m 2 ^ 

 4: dt 



cos (20- 3n't) - \ m 2 <*f- cos (2^ - n'<)l 

 <i dt ' } 



+nV(2~0-r. 



Also the right-hand side of the second differential equation contains the 

 following additional quantity : 



f 21 3 1 



m'nV [48w] \ 3 sin (20 - 2n't) + - e' sin (20 - 3n't) -'-e' sin (20 - n't) [ , 

 I * ^ J 



which, as we shall immediately find, contains non-periodic terms of the order 



,dJ 



y 



282 



