Methodus inveniendi lineas curvas

66 DE METHODO MAX. ET MIN. Conclusio hæc ideo aliquantum difficilior facta est, quod Cycloïs pro recta quacunque instar axis assumpta quæsito satisfaciat, atque æquatio pro axe quocunque admodum fiat intricata. Si autem vel $a$ vel $b$ posuissemus $=0$, quo quidem extensio solutionis non fuisset restricta; æquatio pro Cycloïde statim prodiisset. EXEMPLUM III. 52. Invenire curvam, in qua sit $\int \varrho^n dW$, denotante $\varrho$ radium osculi, & $dW$ elementum curvæ, maximum vel minimum. Per positiones ante factas est $dW=dx\sqrt{(1+pp)}$ & $\varrho=\frac{(1+pp)^{3:2}}{q}$; unde maximi minimive formula erit $\int \frac{(1+pp)^{(3n+1):2}}{q^n}dx$; hincque fit $Z = \frac{(1+pp)^{(3n+1):2}}{q^n}$ & $dZ = \frac{(3n+1)(1+pp)^{(3n-1):2}}{q^n}pdp - \frac{n(1+pp)^{(3n+1):2}}{q^{n+1}}dq$. Quamobrem erit $M=0$, $N=0$, $P=\frac [TRUNCATED]