m L. EULERI OPERA POSTHUMA. Amhjsis. 



Solutio. Sint abscissac datae JF=f, AG=g, quaesitae jP—p et JQ = q^ et in sub- 

 sidium vocetur arcus Je, cujus abscissa AE^ey sitque 



*" *f 1 — eelf " 1 — eepp 



erit \rc.fg — kTC.Ae = — efg[ee-^ff-\-gg — -eeffgg) = M 



' 1 1 



et Arc. pq — Arc. Ae=:-^epq{ee-^pp-i-qq — -^eeppqq) = N, 



ergo Arc. pq — Arc. fg = N — M. 

 Eliminemus autem utrinque e, reperieturque 



gVH-^f*)-fV{i^g^) qV(i-*-p^)-pV{i-*-q*) 



g ■ ■ " ' " ■' — — - , - - * 



i — ffgg i — ppqq 



unde si fgctp dentur, obtinebitur q hoc modo : 



q=[9i^-ffy9-^ffpp-99PP)y{^-*-n{^-*~p')-fi^-ff9g-^99PP-My{^-^9'^^^ 



-+-/>(*— ffpP —99PP -*- ff99)y{^ -*- f) (* -^9') —^f9P {ff-*- 99-*-PP-*- ff99PP)~\ ' 



\i\ —ff99 — ffpP—99PPY—^ff99PP{ff-^99-^PP)\' 



qui valor quoties non flt negativus, praebebit a dato puncto p arcum pq, ab arcu proposito fg 

 geomctrice discrepantem. Q. E. I. . 



65. Coroll. 1. Ambo abscissarum paria ita pendent ab e, ut sit 



ff-^gg = ee{i-*-ffgg)-^2fgV{i-^e') 

 pp-*-qq = ee{\ -\-ppqq) -4- 2pq V^i -{- e*) , 



unde reperietur 



pg (/r-«-gg) - fgjpp^n) . Vii \ c^^i = ^^^ ~*~^^^ ^* -^ff99) — (yy-^-gg) (* -^pp gg) 

 (pq-fg)(i-f9pq) npi-fg){i-fgpq) 



et hinc penitus eiiminando e babebitur 



((1 —ff99) {pp-^qq)-^i^—ppqq) [ff-^99)f= H^-fgpq)'' {(pq—fg^^^-^ilf-^gg) ipp-^qq))> 

 vel ((1 —ffgg)[pp-^qq) — {i—ppqq){ff-^gg)Y= Hpq—fg)'' ((i —fgpqT^-^iff-^gg^^pp-^qq)). 



66. Coroll. a. Hinc ergo dato quocunque arcu fg, inOnitis modis alii determinari possunt 

 arcus pq, quorum differentia ab illo fg sit geometrice assignabilis. Erit autem baec differentia 



Arc.pq — Arc.fg = -^e{ee{pq—fg) (1 ^jppqq—jfgpq^-ffgg) -t-pq [pp-+-qq) ^fg{ff.^gg)) 



e (pq — fg) {ff-*- 99 ■+- pp-*-qq —\pq ipq-*-^fg) (ff-t-gg) — j fgifg-*~^pq){pp-*-^q)) ^ 



~~ ^(i-fgpq) ' . ■ 



67. Copoll. 3. Casus hic duo peculiares considerandi occurrunt, B\ter quo pq = fg , alter quo 

 fgpq=i. Priori casu flt pp -*- qq = ff -*- gg , ideoque p = fctq=g; ita ut arcus pq in ipsum 

 arcum fg incidat. eorumque differentia flat = 0. Alter6 vero casu flt 



