Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 77 CASUS I. 63. Sit in maximi minimive formula $\int Zdx$ terminus $N=0$; ita ut fit $dZ = Mdx + Pdp + Qdq + Rdr + Sds + \&c.$ Hoc ergo casu aequatio pro curva erit haec, $0 = -\frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \frac{d^5T}{dx^5} + \&c.$ quae, per $dx$ multiplicata, fit integrabilis, prodibitque $0 = A - P + \frac{dQ}{dx} - \frac{ddR}{dx^2} + \frac{d^3S}{dx^3} - \frac{d^4T}{dx^4} + \&c.$ CASUS II. 64. Sit & $N=0$ & $P=0$, ita ut fit $dZ = Mdx + Qdq + Rdr + Sds + Tdt + \&c.$ Quoniam est $N=0$, una integratio jam successit, habeturque pro curva quaesita ista aequatio modo inventa, posito etiam $P=0$: $0 = A + \frac{dQ}{dx} - \frac{ddR}{dx^2} + \frac{d^3S}{dx^3} - \frac{d^4T}{dx^4} + \&c.$ quae per $dx$ multiplicata denuo integrari poterit, eritque $0 = Ax - B + Q - \frac{dR}{dx} + \frac{ddS}{dx^2} - \frac{d^3T}{dx^3} + \&c.$ CASUS III. 65. Si fuerit & $N=0$ & $P=0$ & $Q=0$, ita ut fit $dZ = Mdx + Rdr + Sds + Tdt + \&c.$ Bini valores $N$ & $P$ evanescentes jam praebuerunt hanc aequationem bis integratam $0 = Ax - B + Q - \frac{dR}{dx} + \frac{ddS}{dx^2} - \frac{d^3T}{dx^3} + \&c.$ in qua si ponatur $Q=0$, & multiplicetur per $dx$, obtinebitur sequens aequatio ter integrata: $0 = Ax^2 - Bx + C - R + \frac{dS}{dx} - \frac{ddT}{dx^2} + \&c.$ K 3 Ex