A^QVAT, DIFFERENIIALIFM 171 



aa - —1 



a, pj Yj ^"i repericnair cfle ar=^si.-T > Pm-jozrs-i 



I ■ rt ^y ZZdz 



Y— 1T-K2 » Q.uare ncquatio noitra — -+-— f 



y z ^azz—Z—t 



-o,fcu'y H-,^^-|-f;.-i-g-^ (fubftitutis valoribus 

 ipfaruma, (3, v &deinringulis terminis in2rt^-i2 dudtis) 



abibit in nanc aequationem —3; -H -iZiia — -«:^ 



^ -^ir^— <?,in difFerentialibus logarithmicis exprelTam, 

 qux integrata, ut olim docuimus , reddit {2aa—2yj-{-2aa 

 I(z-{-a)-{a-\-iy{z-{-i}-\-{a-i)I[z-i)zzlC, ubi per /Cintel- 

 ligo log irithmnm quantitatis conftantis pro lubitu affum- 

 tx ; reducendo igitur , ut moris eft , logarithmos ad po- 

 tentias, acquiritur aequatio finita feu in terminis finitis ex- 

 prelTa ji=«»-=)x (^+tff «^(zH-i)^-"-'^ (z-ir-'>=:C, 

 Nunc vero, ut in coordinatis x Szy exprimatur , refti- 

 luendus cft valor ipfius z , qui ex hypothefi alTumta xn: 

 jV^zz—i)^ eft iziV{xr-\-yy) : y, hinc enim emergity""*-^^ 



— C ; vel quia in denominatoribus habetur j elevata ad 

 Qaay ad-^— i , & ad^— i, quarum fumma =12^^—25 P^- 

 tet tres iftos denominatores j deftrui per alteramj fra- 

 dionibus prsemiflam , ita ut tandem hacc prodeat aequa- 

 tio naturam curvse determinans {V{xx-]--yj')-^ay)'^'^'^ 

 y{V{xx-{-2y)-\-j)^-''~'^x{V^xx-\-jy)-y}''-'^z=iC ; Qu« fi 

 dextre tradletur ulterius reduci poteft in iftam x^^~'* 

 ^{V^xx-i-xyy-^-ayyx^V^xx-^-XY)-^-^)''^—^ •, vel ctiam in 

 hmcx^-^^-^^V^xx-^-yy^-hayf^x^V^xx-hjy^-yf—OXJbi 

 recordandum, per litteram C intelligi perpetuo conftan- 

 tem arbitrariam in omnibus aequationibus fumendam vel 

 eandem vel diverlam prout libuerit •, quod in fequentibu» 

 ctiam (icubi reperietur monitum volo. 



Y 2 Sin- 



