== (i70 == 



Integratio 

 Aequationis homogeneae 2^ ordinis 



§. 19. Si hic loco dx et dy fcribatur x et jk, denc» 

 minator aequationem fponte integrabilem reddens erit 



a x' -{- (b -{- d) x^j -1- (^ 'h O «^y^ -r-fy\ 

 ita Yt nunc habeamus 



V 3 X axx-hlxy + cyy ^ 



'y x-f-o^y ax' -i-{b-hd]x^ y -i- [c -h e', x y +j y^ ' 



Statuatur iam ifta fradio: 



axx-i-bxy-i-cyy — « 1 W 



ax'-t-{b-i- d)x^ y-^{c-he)x y"" H-J y^ x H- % y 



denotante R omnes reliquas fradiones partiales , vnde multi- 

 plicando per x -H ^j habebimus 



{axx-hbxy-*-cyy){x-h^y) ^ i {^ r^ -\- 5( y) , 



ax^ -i-{b -h d')x^ y -t-{c -h e)x y^ -hf y^ ^ 



vnde pofito x — — 2i y , ex parte dextra fola quantitas a rc- 

 manet, e finiftra vero parte tam numerator quam. denominator 

 euanefcunt. Sumantur ergo differentialia more folito, et habi- 

 ta quantitate y pro conftante prodit ifta fracftio; 



axx-i-bxy-i-cyy-h{x-^-fliy){iax-hby) 

 zaxx-h^^ib-i-d^xy-i-ic-t-e^yy ' 



quae pofito jc — — ■ ^j praebet 



^ a^^ — b^-hc 



30 31^ — a(6_+_di2t-t-ic-+-e) 



Eodem modo porro reperitur quoque 



Q -—- g ^^ 6 Sg -4- C 



* 3 a^2 — 2(b-hd)^-i-{c-he)^ 



— - g £» — 6 g -H c ^ 



( 3 a £^ — 2(6-(-d )€-(-(c -t-e) * 



Cobfficientes autem 2i, 25, C ex fequentibus determinationibus 

 deriuantur: 



y a S{-»- 



