•>l^.l ) 45 ( ^'^<» 



Si igltur hic loco Hterarum E,D,C, fcribamus A,B, C, 

 prodibit aequatio difFerentialis fupra tradata 



d X dx 



y (A -+- iJ X -r- C X jc) V(A -+- B> -(- Lyy) 



cuius ergo integrale erit 



I A-f-B(x-f-y) -t-iCJ7-(-»V(A-t-Bx-f-Cx«j (A - t-B^v-f-Cyjj' _ 



B fe -t- I A -I- I V A (A -»- B <: -+- C il? fe) 

 k k ' 



quae egregie conuenit cum ea in Coroll. I. data. 



Corollarium IV, 



$. s6. Contemplcmur iiunc etiam cafum, quo 

 formula A-i-Bx-^-Cxx-i-Dx^-i-Ex* fit quadratum, 

 quod fit (a -{- b X -i- c X x)% ita vt iam habeamus 



A — aa, B—2.ab, C — bb-^-iaCy D — zbc, E — cc^ 

 tum vero 



y X~ a-\-b x-{-c X X, VY-a-^-by-^- cyy, 

 yKzza-\- b k-^ ckk 

 atque acquatio difFerentialis pro priore cafu erit 



d X djy _ 



a-i-bx-t-cxx a -i- t>y -hcjy ' 



cuius ergo integrale erit 



(^2aa-r-2ab(x +j) + 2 [b b -h 2. a c) xy -{- 2. b c xy (x -1-/) 

 -h 2 ccxxjy -h 2 {a-\-bx-\-cxx) {a-\-by -\-cyj)): 

 {x-y)^ =:A, 

 quae reducitur ad 



aa-i- ah(x-t-y)-i-{'>'--i-ac'ixy-i-^cxy(x-^y)-i-cexx y y 



{X — y]^ ~ 



""'^k ^ ^ ' Quod fi iam vtrinque addamus \ b b, 

 prodibit 



{a + \b{x -f .y ) -f- c xyy —{±±l±!^l_ 

 [x-yY k' 



F 3 vnde 



