Methodus inveniendi lineas curvas

54 DE METHODO MAX. ET MIN. ideoque $\frac{x}{y} = \frac{k+p}{1-kp} = \frac{kdx+dy}{dx-kdy}$, feu $xdx - kxdy = kydx + ydy$; quæ integrata præbet $x^2 - y^2 = 2kxy + C$; quæ est æquatio pro Hyperbola æquilatera. II. Sit $n=1$, & $2n=2$; erit $2A \text{ tang. } \frac{x}{y} = A \text{ tang. } \frac{k+p}{1-kp}$; unde fit $\frac{k+p}{1-kp}$, feu $A \text{ tang. } \frac{2xy}{yy-xx} = A \text{ tang. } \frac{k+p}{1-kp}$; unde fit $\frac{2xy}{yy-xx} = \frac{kdx+dy}{dx-kdy}$, feu $2xydx - 2kxydy = kyydx - kxxdx + yydy - xxdy$; quæ integrata dat $yx^2 = ky^2x - \frac{1}{2}kx^3 + \frac{1}{2}y^3 + C$, sive $y^3 + 3ky^2x - 3yx^2 - kx^3 = C$. III [TRUNCATED]