468 L. EULERI OPERA POSTHUMA. Amiym 



IX. 



Porro vero esry(E-H^~-i-^a?yr = (-5- iy-*-^) h- -ct^") "/(«-»- 2a)-+-( -5- (y— 5) — ^^«)y«— 2a), 



unde colligitur dx^yx-i-Sy-i-^xyy)^ 



f^(r-»-5-H^«)(r-.5~^«)-.-^^(r-a-^u)(r-5~^«)T/;;^ 



<^u dx{yx-t-dy-i-^xyy)=^y^^^^^{y8tt-*-y^u-*-(yy-^dd)U'+'2d^uu-¥'^^u^), 



et quia d;y {yy -i-Sx-t- ^xxy) = — dx {yx -¥-dy-i-^ xyy) , erit 



Vp yV(t* — 4uu) ^ y J V(t* — 4uu) 



X. 



Ut haec formula evadat integrabilis, oportet pro n scribi numerum parem, ut etiam usus hujus 

 formae plerumque exigit. Quare 



si /1 = 0, erit x^ — y^=0 r=Const. 



/1 = 2 x^—y^==y{t'--kuu) . . F=^^ 



n = k x^^f'=tty{t''—%uu) F=^^fttdu 



-Vp 



7 

 — Vp 



7 



etc. etc. 



/1=6 x^'^y^={t' — uu)y{t'---kuu) F=^f{t' — uu)du 



/1 = 8 ««— j«=(«6— 2««MM)y(«*— Jt«w) F=^ f{t^—2ttuu)du 



XI. 



r, .^ ., — a — 25« — tuu 



Lum vero sit tt = ? er 



7 



iuu ^u^ 

 y r 3y 



fttdu = — ^ 



J y . . 



p.. . , aa 2a5 (4S8 -t-^a^ - yy) 3 «5 4 55 5 



/ («* — uu) du = — U-t Mtt -f- ^^ '- — - a^ -»- — tt* -I- ^ tt*. 



J \ ) yf yy 3yy yy ^yy 



Ex his introductis litteris majusculis A, C, E una cum constanti arbitraria c, aequatio in fine art. VII 

 data satisfaciel huic aequationi integrali 



/dx(%-¥-^xx-i--(§:x*) f dy C^ -t- (Syy -*- (§y*) /-. . ^exy (Scxy , Wa -i- Ccc -t- Ec^ Ecexxyy. 



V(A-*-Cxx-*-i:x*) J V(A-t-Cyy^Ey*j' — ^^^^^- ~ VJ ""TT \^^ - ^ ^ ^ »" —37-)- 



Unde sequentes curvarum comparationes adipiscimur. 



Comparatio arcuum Ellipsis* 



I. Expressio simplicissima ad hoc genus pertinens esl utique curva k»mniscata, scd quia com- 

 pnrationem arcuum ejus jam satis prolixe sum pcrsecutus, hic statim ab cllipsi incipiam. Sit igitur 



