1 



330 L. EULERI OPERA POSTHUMA. Astron.mech. 



Potest autem hoc casu acquatio proposita dd& -t- mdu^ sia 2 (« — i9^) =i absolute integ^rari, si multi- 

 plicetur per 2 {du — dO-) , ut sit 



2dadd& — 2d&dd& -h Smdfa^ {du — di'>) sm2{u — &) = 0, ^M 



erit enim 2dud& — d&'^ = Cdu'^ -^mdu^ cos 2 {u> — »^), vel posito u — & = $, sqm d^ = u — s, 

 habebitur du"^ — ds^ =C du^ -^mdu'^ cos2s, seu ds^=du^ (a — mcos25), hincque du = -y- -t-.> 



' ^ ^ ^ y {a — tn cos 2 ») 



ubi a est constans a motu axis ipsi primum impresso pendens. Quoniam igitur assumimus, si 

 momentum vis solis, scu littera./w evanescat, axem esse quieturum, posito /w = 0, crit ds = du, 



ideoque ce = 1 , ita ut sit du = , _ — , ex qua aequatione promotionem axis a vi solis oriun- 



dam definiri oportet. Cum jam sit m fractio valde parva, erit 



1 -i-^Mi cos 2s-i-^m^ cos'^ 2s-*- ^'.'^ m^ cos^ 2^-f- ^ .'V .j n^ cos* 2s-\- etc. 



1/(1 —mcosS») 2 2.4 2.4.6 2.4.6.8 



Potestatibus autem cos25 ad cosinus angulorum multiplorum reductis, fiet 



' 4 * o 1 1..3 2 r * 4-3.5 3 - 1 1.3.5.7 , „ 



= -f-l -♦-— 'wcos^^ -f- ;r-z— ,m C0S4 5-H v-^T— r-7.w^cos65-»- — . ^ . ^ , //rcos85 



/(1 — mcos2«) 2 2 2.4 4 2.4.6 ^""""^ 8*2.4.6.8 



1 1.3 « 3 1.3.5 o 4 1.3.5.7 , 



-*-^-2:4''* -*-4-2:4:6''^ -*-8*2X6:8"* 



3 1.3.5.7 . 



-^¥*2:4-:6:8'" 



etc. 

 Integrando ergo habebitur 'f- \ 



5 35 



-Hg^w^sinG^-f-^m^sinS^, 



4 



rejiciantur termini, in quibus m ultra duas oblinet dimensioncs, eritque 



3 3 1 1 



u = ff-i-u — &-\- jgm^M — le"**'^ -*- T''* ^^" ^ (" — j^) H- — m* sin 4 (a — »9^) , 

 seu i9^ = 5r-»-^m^tt-»--^msin2(a — j^) -i-^m^ sin li- (a — .^), 



axis ergo durante quavis solis revolutione modo progredietur, modo regredietur per arcum = — m; 



1 1 



ita ut si m = g— , hoc spatium futurum sit = — - = 0", l?h = 8' 2V'. Tum vero qualibet revolutione 

 solis, seu singulis annis, axis in ecliptica progredietur per spatium ='—n^. 360®, quod ergo, si 



1 .^ 3.360» „„ 



''' = 2T0' ^^'* = 16:40005 =-^- 41 



Alitcr autem res se habebit, si axis terrae ad eclipticam fucrit inclinatus; tum cnim posita 



orbita solis circulari, ut sit n = et v = u, manente & = u — 5, hae duo habebuntur aequationes 



resolvendae 



dds = 2{ds — du) dcp tang (p -*- mdu"^ sin 2s, 



^ -t- {du — dsY = mdu^ {\ -t- cos 2*). t 



sin f cos q> 



