112 NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. [3 



where 



+ cos 



Cv/i' 77 TO I "^ 



,, on ,dl _ ,., o . 



7 = ^{ 2 ^ +2W 1 2 Sm2w J} ; 



/ 



the coefficients of - , , 81, 8<a, and their differential coefficients being known 

 as functions of (n n') t. 



Now these equations are of the form discussed in Lecture VII., and we 

 may approach their solution in the way that is shewn there ; or still more 

 simply, let P, and (), be the most important parts of X and Y, and c 

 the constant part of /xe~ 3 ', and determine 8,1, 8,^ from the equations 



Then let 



- (l -~- 



- 



cos 



, = A - P, + 2 -2~-n d - 3 (/xe- 3 ' - c) 8,1 + 3n sin 20,8,*., 



and repeat the approximation with X lt Y l in place of X, Y; whence 

 finally if S^, 8J, SJ ... 8,0,, S 2 &>, 8 3 w . . . are the successive corrections found, the 

 complete corrections are 



SI =8,1 +8.1 +8 3 l +... 



8(D = S,o) + 8 2 (i> + 8 3 o> + . . . 



We find 



dn' 



X= [- 0-00584,65667.60159,44 

 n L 



1913,50693,01356,20 cos 2(n-n')t 



18,99962,11544,85 cos (n-n')t 



17227,86035,84 cos 6(n-n')t 



151,90731,94 cos 8(n-n')t 



1,32276,48 cos W(n-n') t 



1144,15 cosl2(n-n')t 



9,85 coBl4:(n-n r )i]. 



