116 NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. [3 



Then writing as in Lecture VII. , 



M 



= n + v , fj.e~ =c + iv, 



v, iv consisting of periodic terms alone, we shall obtain the following 

 equations to determine [SZ] and [Sw], 



~, r n _. / \ fc/Sw^l > r rv 71 , . _ r<f T 



= *' + + 2 ~ 2 n + v ~ 3 (c + w} ^ + 3n sin 2w M ' 



~\ n dl\ dSl ~1 _. / \ fc/Sw^l 



\ + 2 dt V dt \ ~ 2 (n + v) Vdt \ ~ 3 (c 



. , 



dt dt + 2 n + '' + 3n cos 



dl 



where 



, / dl\ 

 X, = X a. cos d> ( 1 + 3c + 3iv) a. sin <p 1 2 ,-- 26, cos <p (n + 1') 



r \ / ' \ fl-t I i \ / 



+ &j sin ^ (3n /2 sin 



F, = 7 - b, sin 0(1- 3n /2 cos 2o>) + 6, cos < (2 - ; - ) - 2ft, sin (n + v), 



\ c ^/ 



and the coefficients a, and /^ are to be so determined that [SZ] and [Sw] 

 may contain no terms involving cos < and sin (j> respectively. Now 

 let P 1 and Q 1 represent the terms in X l and 1^ respectively that have 

 the largest coefficients, excluding all terms in cos <f> and sin <j> ; then if 

 8J, SjW be determined by the conditions 



and JT.,, F 2 be the results of substituting 8^ and SjW instead of [SZ] 

 and [8w] in the right-hand members of the equations that determine 

 and [8w], we shall have as before 



X, = X, - P, + 2 - 2v - - 3^8^ + 3n' 2 sin 2o 8^, 



