Prof. Challis on the Theory of the Moon's Motion. 377 



Third Approximation. 



Small quantities of the fourth order are now to be included. 

 Hence from the complete values of r and </> given by the second 

 approximation, it is required to obtain values of r 2 cos 2</> and 

 f sin 2cf> to the second order of small quantities for substitution 

 in the right-hand sides of the equations (A) and (B). By the 

 ordinary processes of approximation the following results will be 

 obtained : 



^2 cos 2</>= cos 2q + e(cos(2j +p) — Bcos(2q — />)) 



+ e*y— ^cos2q + cos2 (q+p) + - cos 2(q-p)\ 



/15 45 \ 



+ me \ "8~ C0S ^ ~P} — ~o" cos j° ) 



J 19 3 . \ 

 V "8 + '8~ C0 V 



~a sin 2<f>= sin 2q + e(sin (2q +p) —3 sin(2q — p)) 



+ e i ( — ~sin2q + sm2(q+p) + —sin2(q—p)\ 



, /15 . ,. . 45 . \ 3m 2 . 



me \~8 Sm ' ? ~ P > ~ ~8 Sm ^7 + "8" S1U q ' 

 By substituting for r 2 sin 20 in the equation (B), and then 



after integrating and squaring the result,, replacing — —n' bv its 



dd> 3n' 2 r 



approximate value -+ — — cos 2q, the following equation is ob- 



tained: dj? (h _ \ 2 _ 



df \r n '/ ~ 



3/i'V f /5 \ 5 e 2 



<^ + cos2q + ey-cos(2q+p)-5cos(2q-p))- — cos2(q-p) 



2 

 -re 



( 47 _ 29 n/ , 9 _, .\ 

 ^--g-cos2 ? +^cos2( ? +^)--cos2( ? -;j)j 



(7 25 75 \ 



^cos(2y+^) — 5cos(2g— p) + -^-cos(4q— p) — — cos;;j 



+ ^(~l^(2q-p)-lcos(2q-3p)) 



+m2 (rl + ¥ cos4 0}' 



