Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 153 $= -\frac{dp}{p(1+pp)} - \frac{d\Pi}{\Pi}$; feu $0 = ndu + an^n dx \sqrt{(1+pp)}$ $+\frac{ndp}{p(1+pp)}$. Quæ æquatio, ut eliminetur $\Pi$, conjungenda eft cum hac $d\Pi + an^n dx\sqrt{(1+pp)} = gdx$; unde ftatim fit $0 = ngdx + \frac{\Pi dp}{p(1+pp)}$, & $\Pi = -\frac{ngpdx(1+pp)}{dp}$. Cum igitur curva fuerit inventa, hæc æquatio ftatim præbet celeritatem corporis in quovis curvæ loco. Ponatur $dx = -\frac{tdp}{ng}$, crit $\Pi =$ $pt(1+pp)$ & $d\Pi = pdt(1+pp) + tdp(1+3pp)$; hinc- que obtinebitur ifta æquatio, $pdt(1+pp) + tdp(1+3pp)$ $-\frac{ap^n t^n (1+pp)^n \frac{t}{n} dp}{ng} + \frac{tdp}{n} = 0$, quæ tranfmu- tatur in hanc $\frac{npdt(1+pp) [TRUNCATED]