32 LECTURES ON THE LUNAR THEORY. [LECT. VII. 



then if 87 + 8"l, 8'0 + 8"0 are substituted in the complete equations for 81, 

 80 the residuals become 



X" = 2 ^ C ^- 1 - 2r , - 3iv8"l + 3n k sin 2o,S"0, 

 at at at 



dld8"0 d8"l , , s , ;/) 



y " = 2 -r- 5 \-~2v +3 ir cos 2w8 0, 



dt dt at 



expressions which, if developed in series of cosines and sines of multiples 

 of t//, will have coefficients of higher order than the corresponding coef- 

 ficients in X', Y'. The like process may be repeated until the residuals 

 become insensible ; we then have sensibly correct values of 81, 86, giving 



We may now take into account squares and products of the small quan- 

 tities 81, 80 by treating 1 + 8I, + 80 as given approximate solutions just 

 as we have here treated /, ; substitute them in the complete equations of 

 motion, and determine the residuals X, Y which they leave. These residuals 

 will form the basis of a second approximation, and the operation may 

 be repeated until no further correction is necessary. It is to be observed 

 that if S/, 80 depend upon some such constant as the eccentricity of the 

 Earth's orbit around the Sun, or the parallax of the Sun, then successive 

 approximations yield correctly and separately the terms which depend upon 

 the first, second, powers of that constant. 



