122 THE SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [4 



(log -} = - wVo, cos A' - (2 - 2m) 2 rfa, cos 2 f - ( 2 - 3m) 2 w 2 8 cos ( 2 - <') 

 cfa 2 \ & r] 



- (2 - m) 2 n 2 9 cos (2+ <') 

 + - mV8a 3 - 2mn -^- a sin ft 



+ \ - (2 - 2m) 2 n=8a 5 - 2 (2 - 2m) n ^- 5 - 2a 5 ^ sin 2f 

 + I" - (2 - 3m) 3 n'SOi - 2 (2 - 3m) n -^ - 2a 8 -Jj sin (2f - <') 

 + [ - (2 - m) 2 n*8a, -2(2- m) n ~ - 2a ~ sin (2+ fl) ; 

 ^ = _ mV &, sin </,' - (2 - 2m) 2 n6, sin 2 - (2 - 3m) 2 n 2 6 8 sin (2 - f ) 



c?n f 0^7 c?6.~l .. 



+ -j- + mVS&j + 2mw -T- cos 



+ I" _ ( 2 - 2m) 2 S 86, + 2 (2 - 2m) n ^ + 2& 6 



cos 2 



+ f _ ( 2 - m) 2 n*8b, + 2(2- m) w -^ + 26 S ^1 cos (2f + f ). 



Hence observing that 



1 da _ 2 dn 

 a-di~ ~ Zn~di' 



and substituting in the equations, and further writing 



~ cos <)' + u 5 cos 2f + M 8 cos 2^ ' + J6 9 cos 2 



cos 2 (^ - ff) =p, +p, cos ^' +p t cos 2f +^ 8 cos (2f - f ) +p 9 cos 

 sin 2 (^ - ^) = ^ sin <f>' + q s sin 2f+ ?g sin (2f - </>') + q t sin 



we obtain the following equations by equating to zero the coefficients of 

 the different new terms introduced. 



