34 LECTURES ON THE LUNAR THEORY. [LECT. 



and 



15 9 . 9 /45 15 



Assume 



8 = Xj cos t// + Xa 3 cos 3//, 



80 = A?*! sin // + X7,> 3 sin 3t|/, 



neglecting for the present the terms in 5\jj. In the present case it happens 

 that it is more advantageous to substitute these expressions directly in 

 the complete equations for 81, 80 given in the last lecture than to follow 

 exactly the process for rinding them by successive approximation. Omitting 

 the factor X, we get 



j cos // + 9 3 cos 3i// + 4a, sin 2\jj [a, sin \jj + 3a 3 sin 3\j/] 



+ 3 -^ [X cos i|/ + a, cos 3i//] + 3 ^ 3 [3a., cos 2<//] [a } cos i/ + 3 cos 



-[2(1+ %') + 4k, cos 2i//] [6, cos i// + 3b :) cos 3t//] 

 + 3w'- [sin 2i/ + 6., sin 4i//] [/;, sin i/ + b 3 sin 



cos ^-n- + ~ &.-^, cos 



- 6j sin t/ - 9/> 3 sin 3/ + 4a, sin 2i/ [6, cos $ + 3& s cos 3/] 



+ Sri* [-b, + cos 2i// + b, cos 4V/] [6, sin t/ + 6, sin 

 + [2 (1 + ri) + 46 2 cos 2^r] [a, sin i/r + 3 3 sin 3i/] 



sin ^ +n/2 5 + 6 - ^ 8in 



(45 15 \ 

 b. 2 ~ a, ) sin 5^ = 

 lo ID / 



