LE VERRIER ON THE PERTURBATIONS OF PLANETS. 351 



{i, —i )cos( —i )na.— [i, —i ]sin( —i )??a^ 

 4- {i,—i—l) cos{ — i—l)nct— [e,— i— l]sin( — i— !)??« 

 + {i,—i+l) cos{ — i+l)noi—[i,—i+l]sm{ — i+l)nci . i 





,—i—p) cos{ — i—p)nx— [i, —i—p']&m{ — i—p)nu 

 ,—i+p) cos{ — i+p)na— li,—i+p']sm{ — i+p)nu^ 



— i ]cos( —i )««+ (i, —i )sin( — z )nct 

 + [i, — i— l]cos( — i— 1)m«+ (i, — i— l)sin( — i— l)«a 

 + [*>~*+ l]cos( — i+ !)««+ ('<,—«+ l)sin( — ^+ \)not 



>{M.) 



' =B/''\^ 



+ \_i, — i—p']cos{ — i — p)nu+ {i, —i—p) sin{ — i—p)noi 

 + [i,—i+p']cos{ — i + p)noc+ (i,— i+jo)sin( — i+jo)wa, 



I now eliminate successively between these two equations 

 [i, — i], then {i, —i), and I obtain the two relations: 

 (i, — i) +{{i, —i—l) + {i, —i+ 1) } cos «« -, 

 + { [i, — i— 1] — [i, — i+ 1]} sin «a 



4- { {i, —i—p) + {i, —i+P) } cos pnu 

 + { Ih — ^ —p] ~ Ih — ^ +P] \ sin pnoL^ 

 \i, —i] + {{i, — i+1) — {i, —i—l)} sin no. 

 + { \i, —i—l] + [i, — i+ 1]} cos ncK. 





\=QLf"', 



(34.), 



(35. 



(36.) 



+ { (i, —i-\-p) — [i, —i—p) } sin jowa 



+ {[i, —i—p'] + [i, — i+jo]} cospnot. 

 in which I suppose that P/"^ and Q/"^ have the following 

 values : 



P/")=A/") cosi??« — B/"^ sin inu,'] 



Q/") = A/"^ sinina + B/"' cos iw«. J 

 Let us, to simplify the matter, put 



(i, —i—p) + (i, — i+Jo) =^p, 

 [i, -i-/.] - [i, -i +^] =r^ ; 

 the formula (34.) will become 



(i, — i)+a;'iCOS nct + Xc^cos 2na. + . . . + Xp cos pnc^'^ 



+ s:^ sin w « + a-^ sin 2 « a + . . . -\-Zp sinp nuj~ ' 

 and by attributing to n the different entire and positive values 

 on starting from zero, we shall have a system of equations 

 similar to the system (4.). We shall then deduce from them in 

 the same manner the values of (i, — i), Xp and z,,. 



