Methodus inveniendi lineas curvas

ELASTICIS. 271 $\int \frac{qqdS}{(1+pp)^{\frac{3}{2}}}$; fitque H ejus valor, si ponatur $x=a$, cujus quidem constantis $a$ consideratio mox ex calculo rursus evanescet. Erit ergo $V = H - \int \frac{qqdS}{(1+pp)^{\frac{3}{2}}}$: Unde valor differentialis fiet $= -\frac{dP}{dx} - \frac{1}{dx}d.[P]V + \frac{ddQ}{dx^2}$. Quamobrem ex his duobus valoribus differentialibus nascetur haec aequatio pro curva quaesita $\frac{a}{dx}d.\frac{p}{\sqrt{(1+pp)}} = +\frac{dP}{dx} + \frac{1}{dx}d.[P]V - \frac{ddQ}{dx^2}$, quae integrata dat $\frac{ap}{\sqrt{(1+pp)}} + \mathfrak{C} = P + [P]V - \frac{dQ}{dx}$, five $\frac{ap}{\sqrt{(1+pp)}} + \mathfrak{C} = \frac{Hp}{\sqrt{(1+pp)}} - \frac{p}{\sqrt{(1+pp)}}\int\frac{qqdS}{(1+pp)^{\frac{3}{2}}} + P - \frac{dQ}{dx}$, ubi constans H alias determinata in constante arbitraria $\mathfrak{a}$ comprehendi potest, quo ipso constans $a$ ex calculo egreditur. Idcirco ergo prodibit haec aequatio $\frac{ap}{\sqrt{(1+pp)}} + \mathfrak{C} = P - \frac{dQ}{dx} - \frac{p}{\sqrt{(1+pp)}}\int\frac{qqdS}{(1+pp)^{\frac{3}{2}}}$. 43. Multiplicetur haec aequatio per $dp=qdx$, atque prodibit: $\frac{apdp}{\sqrt{(1+pp)}} + \mathfrak{C}dp = Pdp - Qdq - \frac{pdp}{\sqrt{(1+pp)}}\int\frac{qqdS}{(1+pp)^{\frac{3}{2}}}$. Cum autem fit $dz = \frac{qqdS}{(1+pp)^{\frac{5}{2}}} + Pdp + Qdq$, erit $Pdp = dZ - Qdq - \frac{qqdS}{(1+pp)^{\frac{5}{2}}}$; quo valore substituto emerget aequatio integrabilis haec: $\frac{apdp}{\sqrt{(1+pp)}} + \mathfrak{C}dp = dZ - qdQ - Qdq - \frac{qqdS}{(1+pp)^{\frac{5}{2}}} - \frac{pdp}{\sqrt{(1+pp)}}\int\frac{qqdS}{(1+pp)^{\frac{3}{2}}}$, cujus integralis est: $a\sqrt{(1+pp)} + \mathfrak{C}p + \mathfrak{y} = Z - Qq - \sqrt{(1+pp)}\int\frac{qqdS}{(1+pp)^{\frac{3}{2}}}$ seu