25i L. EULERI OPERA POSTHUMA. Astron.mch 



qq = kk-i-7if-*-^{i—kk)-i-' -^ 1 h-^ -^ -^H 5^ ^. 



Tum autem formula irrationalis induit hanc formam 



qsins^//jt q ~, D(3-i-ico8*) £(6 -»-4ft co8»h-M(1 -*-co8^»)) 



— y(^— 6— (5 — 



F(10-t-10A:cos«-t-5fcfe(l -♦- co82»)-h*'cos«(1 -hcos*»)) 

 _ 



/r 



unde concludimus 



dv qdfsins kdtpiins //^ D (S -*- k cos s) £(6-4-4A;co8»-t- AA:(l -♦- cos'»)) . \ 



— = ^r^ ( ^ H 1 :; -*- etc. )• 



_ . ^. dv qdssins dp (p^ — Qdp) j ..^v/o^^n 



Est vero etiam — = 1 —^ — ^-^ cos s, unde 



vv p pp pp 



osins,, ,\ kd(psins//Y Z)(3-*-fccos*) £(6-»-4tcos»-*-W(l-f-cos*»)) F/10-l-10*cos«-»-W(5-*-A:cos»)(l-i-cos*»1)\ 

 L-(<l9P-*) = -^(S-H p -K- j .'+ —^5 >2) 



miol oaiid JJtisitcd tmium fM 



dp (pdq — jdp) cos» 

 "*~^ pF ' .Jlfi|_ IKf =rq Olfli/01 



ubi quidem haec differentialia ipsarum dp et dq non tam commode exprimere licet, quam ante. 

 Quoties autem unico termino constat mte^rale f(T^'d(p -^ Fdv) , toties posterius membrum reduci 

 potest ad formam per /csin^ multiplicat^^. Est autem in genere ' ,lnvr,^ ,Mt^:' 



'.*'.' t ti' i<> ( 



dp (pdq — qdp) cos « —Tv^df.^ , . Wi \ d^cos» d35 (i-»- co««) d@ (2*-»- (1 -♦-**) cosiVi 



jp —^{^ii-^-[i-^-mcoss) ^^ 2A ' — Wk 



dD(3*-*-*^-+-(l-^3M)cos») d£(4A:-H4A3-*-(l-*-6**-i-*4)cos») dF(5*-+-10A3-^-ii:S-»-(l-*-lO**-»-5Jt*)co»») 



' etc» 



1 



^r^k ^f*k 2rn 



quae exnressio transmutatur in hanc formam ..... 



^ '•nBft fli Jidfi oilBupec lonq i 



— (2/c-*-(l-*-**)cos»)w /^ j ,„, d35 d@ dD dE dF ^ \ 



\ V f V fvv ^ ' ffvv ^ V ' 



(AA-^|)/9-f -^! 



f^vv ^ V vv ' 



At ex aequatione assumta est 



d%-{ 1 1 — r-t-etc. = — Tvdw — Vdv -\ H-^=V-etc. 



V VV V'' ' VV v^ 



ita ut prius memhrum superioris aequationis aheat in 



"2^ 



(2*-f-(l-4-M)cos») , / S 26 3D AE 5F \ 



(.n~ 2/r* vvt*»'\^'' ^^ ^3 ^4 ^s ^6 ^''*'-; 



