QVIBFSD. CALCVLl INTEGRALIS. 251 



Hefccre. Ex hac vkima aequatione veio deriuatnr 

 «^.v =: l^, q"ie integrata haec eft , .r = /(2 -+- «), 

 vel, pofito /t?— I5 crit x/ezz^ l {Z -\~ n) , aut vero 



zz 2 H- «. 



§, 18. Ex hoc artificio apcritur campus varfas 

 -aequationes, difficiles alias apparituras, ad log.irithmos. re- 

 ducendi , quem vnico exemplo coromonrtrabo : fit 

 dx ■+■ dj ~ [bdx -f- -2 cxdx + 3 ex^dx -}- ^f/yH- 2 bjdy 

 H- 3 ky^dj) : (<z -f- ^.v -f- cx^- -H ex' -^- j -^ £}' + /y' 

 -f- ky^) , quod efl: -diffcicntiale Logarithmicus , cuius 

 nempe numerator elt differentiale nominatoris in hac 

 fi-adione, ita vt integrale huius acquationis fit x -^ y 

 ~ log. [a -i-bx-h c x^- -i-e.x' -^f-^-gy -h hf -f- ky') ; 

 muinplicetur dx -\-4y per nominatorcm fmdionis , re- 

 fultabit exinde fcquens aequatio, ad logarithmos reducenda: 

 {a-\-f- b) dx -V- {b- 2c)xdx -]- [c - :^e)x'dx 

 H- ex'dx -\-9jdx -i- hyUx -|- ky^dx ^-i^(^a-^f—g) 

 dy -\- bxdy -i- cx^-dy -\- ex^dy + {>^- 2.h)ydy + {b-ik) 

 y-dy -\- ky^dy -zzo^ quod paradigma ipfum iam ad caltis 

 pkr.imos integrandos , fine pracuia inacterminatorum 

 feparatione inferuire poteft. Sit ex. gr. hoc modo iii- 

 tegrandum 3J^a,' — sy'dx — 3 dy -j^ ^y dy -\- i sy^dy 

 — sy^dy rr 0. Obferuabitur ex comprtrationc termino- 

 rnm homogeneorura, > fll g -:. 3 , (■ _- — <f , f— — a, 

 h ~ o^ b ~ ^ : :z: o^ e ■::: . . ..itegrale 



erit hoc, x H-j n: log '^l^y • 



;cp. 



