26 LECTURES ON THE LUNAR THEORY. [LECT. 



so that 



r ( dt \-] f (^\_ l i r i 11 :} 



but 



log, in T) = - (1 - m)" 6 2 2 + (2 - 2m) b. cos (2 - 2m) 



+ [(4 - 4m) b t - (1 - m) 2 6 2 2 ] cos (4 - 4m) 0, 

 log e au - j- + cos (2 2m) 6 + ia t -j-j cos (4 4m) 6. 



Hence we find 



a\*> 7 " 1 / ' t '^ v l i * > 



-m)-l>.; = \og e (-^\ + -a.;, 



(2 -2m) b.,= -k,~2a,, 

 (4 4m) 64 ( 1 m) 2 b.? = li t 2a 4 + - a, 2 . 



The remaining equations of condition that we require are obtained 

 from the first equation of motion ; this may be written 



*(u) , 



Now 



an = 1 + a, cos (2 2m) 6 + a t cos (4 4m) 0, 

 whence 



' ^' = - (2 - 2m) a,, sin (2 - 2m) (4 4m) a 4 sin (4 4m) 0, 



^72 l,\ 



= - (2 - 2m) 2 , cos (2 - 2m) - (4 - 4m) 2 a t cos (4 - 4m) 0, 



and 



= m 2 [1 + (4 - 4m) 6, cos (2 - 2m) 0], 



In' -^\ sin 2 (0 - 0') = m 2 [sin (2 - 2m) + (2 - 3m) b, sin (4 - 4m) 0], 



