Methodus inveniendi lineas curvas

128 DE METHODO MAX. ET MIN. minari quantitas $n$, hocque pacto inveniri æquatio pro curva quæsita. Hoc autem modo calculus fieret maxime tædiosus, ac minime tractabilis. Adminiculum vero summum afferet ultima æquatio in hanc formam transmutata: $\frac{Cdp}{gp^3} = \frac{(n+\frac{1}{2})dx\sqrt{(1+pp)}}{n\sqrt{n}}$ $-\frac{nCdx(1+pp)}{np}$, cui expressioni ante æqualis esse inventus est valor ipfius $dV$; erit ergo $dV = \frac{Cdp}{gpp} \& V = D - \frac{C\sqrt{(1+pp)}}{gp}$. Jam igitur habemus duas æquationes has $\frac{Cdp}{gp^3} = \frac{(n+\frac{1}{2})dx\sqrt{(1+pp)}}{n\sqrt{n}} - \frac{nCdx(1+pp)}{np}, \& D - \frac{\alpha C}{gp} = \frac{1}{n\sqrt{n}} - \frac{C\sqrt{(1+pp)}}{np}$. Ex quibus si eliminetur $n$, habebitur æquatio inter $p$ & $x$ ejus- modi, ut nusquam $x$ sed ubique [TRUNCATED]