Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 59 nam gradus futura fit æquatio differentialis resultans. Accidit autem hoc, si maximi minimive formula proposita fuerit hujusmodi $\int Zqdx$, existente $Z$ functione tantum ipsarum $x$, $y$ & $p$: ita ut sit $dZ=Mdx+Ndy+Pdp$. Hinc igitur erit $d.Zq=Mqdx+Nqdy+Pqdp+Zdq$: unde pro curva quæsita orietur æquatio hæc $0=Nq - \frac{Pdq+qdP}{dx}$ $+\frac{dMdx+dNdy+Nddy+Pddp+dPdp}{dx^2}$, seu $0=2Nq$ $+\frac{dM+pdN}{dx}$, vel $0=2Ndp+dM+pdN$: quæ æquipollet tantum æquationi differentialis secundi gradus, propter $dp=\frac{ddy}{dx}$ quod inest. Si igitur curva desideretur, in qua fit $\int Zqdx$ maximum vel minimum, existente $Z$ functione ipsarum $x$, $y$ & $p$, atque $dZ=Mdx+Ndy+Pdp$; pro curva quæsita habebitur æquatio $0=dM [TRUNCATED]