Methodus inveniendi lineas curvas

68 DE METHODO MAX. ET MIN. valor litteræ D tribuatur, etiamfi alia æquatio inter $x$ & $y$ proveniat; verumtamen ad alium axem relata. Notare interim convenit pluribus cafibus curvam algebraicam fatisfacere; quorum quafi primus eft fit $n = -\frac{1}{2}$, quo erit $x = \int \frac{adp}{(1+pp)^{3/2}} = \frac{ap}{(1+pp)^{1/2}}$. & $y = \int \frac{apdp}{(1+pp)^{3/2}} = -\frac{a}{(1+pp)^{1/2}}$; unde fiet $(1+pp)^{3/2} = -\frac{a}{3y}$ & $pp = \sqrt[3]{\frac{aa}{9yy}} - 1$, quibus fubftitutis refultat $x = (2\sqrt[3]{\frac{aay}{9}} + y)\sqrt{(\sqrt[3]{\frac{aa}{9yy}}-1)}$, æquatio algebraica pro curva, cafu quo eft $n = \frac{1}{2}$. EXEMPLUM IV. 53. Invenire curvam, in qua fit valor hujus formulæ $\int \frac{ydydx}{ddy}$ omnium minimus. Patet primo maximum locum habere non poffe, quia in linea recta fit $ddy=0$; [TRUNCATED]