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



give rise to terms of the form ~M cos 2 (q—p) in the value of 

 r 2 sin2</>. Also a term of the form B cos 2 ((7— p), and of the 

 third order in r, will produce a term of the same form and of the 

 second order in the expansion of 6. Let us, therefore, suppose 

 the values of r and (f>, which are to be substituted in the small 

 terms of the equations (A) and (B) for a second approximation, 

 to be the following : 



-=1— ecos»+Acos2p + B cos2(q — p) 

 a 



if) = q 4- 2e sin p + A' sin 2p. 



On going through the calculation, and retaining terms of the 

 fourth order which involve the argument q—p, the following 

 result is found for the equation C : 



dr 2 A 2 2/t w!*r* p _ 

 <ft 2+ r 2 r 2 + 



ra' 2 a 2 -< 3<? cos (2q— p) — e cos (2q +p) + 



In order to integrate this equation, assume 



- = 1 — ecos^ + S . Pcos {aq + (3p), 



and substitute in the first side of the equation. Then by making 

 the terms of like argument identical, different corresponding 

 values of P, a, and yS are found, which determine the different 

 terms of the development of r. Those of the development of 6 

 may then be obtained by integration. For the values u — 2, 

 /3= — 1, there results 



/51e 2 3(A-A') 57e 2 \ 



3m 2 /17e 2 A-A' \ 

 4e \4?n m /' 



and as A is the coefficient of cos 2p in the development of r, and 

 A' the coefficient of sin 2p in that of 6, we have 



. e 2 ., 5e 2 , „ 15?n<? 



A=- y , A'=— , and P=--g-. 



The corresponding coefficient in the development of 6 is plainly 



-\ — . It appears also that the coefficient B is t-j\ — > an< ^ 



that the coefficient of the term containing the argument 2(q— p) 

 in the expansion of is reduced to zero. 



