Problemata Astronomiam mechamcam spectaniia. 331 



Multiplicetur acquatio prior per 2qds et posterior per — dq^ ambaeque invicem addantur, erit 



2qdsdds — kqds (ds — du) d(p tang; (p 



JH^l^^dqidu^dsf ' 



8in f cos cp ■* ^ ' - 



et partem posteriorem integrando fiet 



dqddq> ,' , ,.2 \-=2mqd{i^ds^\vi2s — mdv^dq — mdu^dqcos^s 



sin f cos f ^ ^ ■' J ' 



C — mqdu^ — mqdu cos 2* = 



sin f cos f 



// '^ q as aa s — 'i^qas \^as — au) aq) iSin^ q> \ 

 f-^_ _ ] 



^ sin f cos y» ^ \ ' / 



Sit nunc 9 = cos^9?, erit dq — — 2^90 sin^^cos^p, ideoque 



Cdu''--mdu^co%^cp{\-^~co%2s^ = fC'^^^'''''^^'^^^^'^^'^^''^^^ 



^ ^ -' \-^2dcpddq)-+-2(du — dsY d q) sin q) cos q) J 



—f[2dipdd(p ~i-2dsdds cos^ q) — 2ds^d(p sin q) cos (p -+-2du^d(p sin (p cos g)) 



= d(p^ -+• ds^ cos^ ^ — du'^ cos^cp, 



' Quocirca erit Cdu^ = d(p^ -*- (d*^ — dtt^) cos^^? -h mdu^ cos^^d (1 -*- cos 25). 



Si jam sumamus casu, quo /?i = 0, axem quiescere, ut sit ds=du et d5p = 0, fiet C=0 et 



dq) 



= da^ — ds^ — mdu^ (1 -f- cos 2 5) , hincque 



cos 93 



du 



-y^ 



COS*(p 



m (1 -I- cos 2 s) 



I Verum constantem C potius convenit definiri ex statu quopiam axis initiali. Si igitur assumamus 

 principia, ubi axis primum a vi solis comitari coepit, fuisse angulum s = u — d- = s, et inclinatio- 

 nem ^ = /; in hoc statu motum axis nullum statui oportet, seu erit ^1^ = et d(p = Oj ideoque 



j ds==dUf quibus substitulis fiet Cdu^ = mdu^ cos^^y (i -i- cos 2 s)^ unde hanc obtinemus aequationem 



mdu^ cos^y (1 -+- cos 2s) = d(p^ -+• ds^ cos*<jd — du^ cos^^d -h mdu^ cos^ 9^ (* ~*- cos 2*), 

 ex qua oritur 



_l 2 d(p^-\- ds^ cos^ f 



cos^g)-i-m cos^y (1 -t-cos 2f) — m cos^^s^l -♦- cos 2») 



Quoniam inclinatio (p minime a primitiva y discrepat, ponatur (p=y-t-o), erit co quantitas minima, 

 tit d(o prae ds pro evanescente haberi potest. Fiet ergo d(p^=dco et cos^ = cosy — wsin/, 

 atque cos*9? = cos^y — «sin2/, quo valore substituto erit , 



du 



2 doj^ -*- d*'^ cos2 y — <jd«*8in2y 



co8*7 — cjsin 2y-t-mcos 2« -*-mo sin 2 7 — mcos^ycos 24-*-ni(j sin^^cos 2« 



, „ (l«2 d<y2 



Seu du^ = r-;r- ■+ 



, n mcos 2£-Hm6) sin 2v cos*y H-m cos 2£ — « »in 2y — «icos^ycos 2» 



1— mcos2«-i-. : ' 



cos"^ 7 — w sin 2 y 



