Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 95 EXEMPLUM III. 17. Denotet $\Pi$ superficiem solidi rotundi ex conversione curvae ah circa axem AZ orti, quae est ut $\int ydx\sqrt{(1+pp)}$, hujusque superficiei functio sit quacunque $Z$, invenire curvam, in qua pro data abscissa $AZ=a$, maximum minimumve fit $\int Zdx$. Ob $dZ=Ld\Pi$, erit $L$ functio ipsius $\Pi=\int ydx\sqrt{(1+pp)}$; & ob $d\Pi=ydx\sqrt{(1+pp)}$ fiet $[Z]=y\sqrt{(1+pp)}$, & $d[Z]=dy\sqrt{(1+pp)}+\frac{ypdp}{\sqrt{(1+pp)}}$: unde erit $[M]=0$; $[N]=\sqrt{(1+pp)}$; $[P]=\frac{yp}{\sqrt{(1+pp)}}$, reliqui valores $[Q]$, $[R]$, $[S]$, &c. omnes erunt $=0$. Quocirca pro curva quaesita ista habebitur aequatio: $0=(H-\int Ldx)\sqrt{(1+pp)} - \frac{1}{dx}d.\frac{yp}{\sqrt{(1+pp)}}(H-\int Ldx)$. Ponatur, brevitatis gratia, $H-\int Ldx=V$; [TRUNCATED]