J30 DEMONSTRATIO TREOREMATIS 



vnde lequitiir fbre : 



^v '^yx->r^xxy--WV[-ci^-^{yy-aL^ (3(3)rr (3^v*) 

 ^x-{-yy-\-^xyy-±y{'a.^-\-'[yy-v.^-fi^)jy-^5y*) 



Quodfi veio aequatio propofita diffcreniietur, orietur: 

 o~ p vdv-^^ydy-^-yvdx-^-yxdy-^Bxyj^^-i-^xxydy 



feu o-=i:dx^x-\-yj-\-S xyy) -\-dy' (3/ -4- yx-\-6x xy) 



quae abit in hanc : 



<iy I dx — 



^x~i-yy-^-Sxyy 1 ' ^ y -f^y x ~i-S xxy ^' 



Subftituantur loco denomhi;Uorum formulae illae irratio- 

 nalcs, vt prodeant duo membra differentialia, in quibus 

 variabiles x tt y fint a fe inuicem feparatae , ac fu- 

 mendis integraiibus obtinebitur : 



Coroll. r. 



2. Summa harum formultirum integralium erit 

 conftans , fi hi vtraque radicis extradione fignis radica- 

 libus paria tribuantur figna ; fin autem figna ftatuantur 

 dilparia , tum difFerencia fbrmulirum integralium erit 

 conftans. 



Coroll. 2. 



3 Si ponamus : 

 .-a(3:=A^; yy-oL$-p^ — Bk', -p6' — Ck, 

 l-nde fiet : 



Qiiare fi relatio inter Jt et y hac aequatione expri- 

 iratur : 



o — ' Ah-{ p^[xx-\yy)+2xyy{ACkk-{-Hpp \- 13*;~ Ckxxyy 



cric 



