14^ ATTI 



ve! minimum fpatium comprehcndat. la hoc autem probkmate 

 correda ctiam Euleri lequatione, adeo implexus eli calculus , 

 ut nulla redutìionis appareat ratio. Cum enim Eulerus pag. 8i, 



fcripfiflet 6(~ -v~)- ^ locoó^i-^^ j. r p » correftis termfnis 



omnibus qui inde pendcnt, fic a-quario propofìtì problematis 



«dy — hdx H ^ — ( à.dy^-^x di^-^ ^ — ■- H ^ 



idddy . rf/4 N 



C O R O L L A R 1 U M IL 



Si requìratur curva» in qua maxima vel minima flt quan- 

 titasS {x^-^ y"^ )' . ds ì Se invcftigatio hujafmodi transferatur ad 

 elf-mentum {x^-^-y'^ )" ds^ co ad priecedentes formulas relato, fiec 

 Y=2«(:v*-i-j'*)""~*7^j:, erÌEque(;(r'*-t7^j''coefficiens elementi arcus 

 iis ? adeoque prò cafu maximi , aut minimi valons , iiet 



2« ( Ar*-Hy '^)'~^'yds — d (x^^^y-^ \ ^^ ~ ° ' ^^ ^^^^^ P^" 

 fìerior terminus —i n(x^-^-y'*)''^^(xdx-i'ydy — — 



■^ n rt f^'^y dydds\ n f j 1 dyddy 



{x*->ry^) [ — ~~—--r ) , atquc eli mfuper ^^j= — -— . Quare pra 



^ \ds ds-'* ) ' as 



dy 



Cafu maximi , auc minimi valoris , iiet 2 n {xdx-^ydy ) "r "** 



dx^ddy /'ydx—xiy\ 



{x*-i-y^)—jY'—^^ydSi atque inde eruetur 2 7?f — ^-— — 1=: 



1 : quae asquatio ,, pofito dy=^pdx ■, & ds—\/[dx°^-^dy'^ ) — 



ds 



(yd X'^—^ dy\ 

 —z — 4-) — 



/? -fi 



- — — , quam Eulerus exfuis. forra aliscollegerat in exemplo vii, 

 Propof. 111. Gap. IK 



C O R Q I. I. A R l U M III. 



Si in formula propoltra fìat m=^o, & requiraturcurva , in 

 qua fumma omniuni ds lic minima, atque in priori formula 



pcoblematis fiat cocfficiens elementi arcns — = 1 , ent rf( — — 1 



-d 



