210 L. EULERl OPERA POSTHUMA. A^iron.mech. 



. i__y(i _nn) .. 



Ponatur brevitatis gratia = /w, erit 



ds = do (l -+- 2wcos (?-+-2wi^ cos 2o-i-2m^ cos 3(?-»-2ot* cos 4^-4-etc.) 



hincque integrando 



5 = <7 -f- |-m sin <7 -4- -|^ /?i^ sin 2 <7 -f- |- m^ sin 3 <? -I- -f- m^ sin ^^ -H etc. 



Cum nunc sit 6 = t -t-n sin <?, erit aequatio centri 



s — T = (2/w -I- 7i) sin <? H- -| m^ sin 2 <7 -I- |- m^ sin 3 <? -I- ^ m* sin ^ <7 h- etc. 



Potest etiam anomalia media r per veram s simili modo exprimi; cum enim sit 



3 



j ds{{ — nn) 2 



dT = -j, 



3 



erit dr = (1 — m)z ds (1 — 2wcos5-h 3/i^ cos'*^ — kn^ cos^ 5-4-5/1* cos* 5 — etc), 



cujus seriei, si potestates cosinus s ad cosinus multiplorum angulorum revocentur, prodit terminus 

 constans 



1 -h| /i*-i-|^/i* -I- etc. = (1 — /i/i)""^ 



et coefficiens ipsius cos5 Gt 



= — 2/1 (1 — /i/i)""». 

 Quare ponamus 



dT = ds[i — Acoss-^ B cos2s — Ccos 3s-i-Dcosks — etc), 



quae series per (1 -i- /i cos sY == i -h--j' nn -k- 2 n cos s -t- -{ w'» cos 2* multiplicata dat 



1 -I- |-/i/i — A(i -f-^ nn) coss-i-B (1 -t-{ nn) cos 2s — C{i -t-^ nn) cos 3* -*- D (1 -f-^/i/i)cos ^5 etc 

 — Jn -t-2n — Jn -h- Bn — Cn 



-i-\Bnn-t-Bn r-Cn -¥-Dn — En 



^i-Jnn . -i-j-nn ^±.Jnn -*-^Bnn 



— fC/i/i H-fD/i/i — ^Enn -t-fF/i/i ♦ 



3 



aequari debet ipsi (1 — /in)T. At est J = 2n, unde fit primo 



. f .'-L-^-i- j-/m- — 2nn ^ j;- Bnn = {i — /1/1)2", 

 ideoque 



