Methodus inveniendi lineas curvas

116 DE METHODO MAX. ET MIN. $d.[Z]dx = nv.dx ( [L]\alpha + \frac{[Q]}{dx^2} )$ $d.[Z']dx = nv.dx ( [L']\mathfrak{G} + \frac{[P']}{dx} - \frac{2[Q']}{dx^2} )$ $d.[Z'']dx = nv.dx ( [L'']\gamma + [N''] - \frac{[P'']}{dx} + \frac{[Q'']}{dx^2} )$ $d.[Z''']dx = nv.dx.[L''']\delta$ $d.[Z^{iv}]dx = nv.dx.[L^{iv}]\epsilon$ $d.[Z^v]dx = nv.dx.[L^v]\zeta$ &c. Ex his igitur erit ut sequitur $d\Pi = \alpha$ $d\Pi' = nv.dx ( [L]\alpha + \frac{[Q]}{dx^2} )$ $d\Pi'' = nv.dx. ( [L]\alpha + [L']\mathfrak{G} + \frac{[P']}{dx} - \frac{[Q]+2d[Q]}{dx^2} )$ $d\Pi''' = nv.dx ( [L]\alpha + [L']\mathfrak{G} + [L'']\gamma + [N''] - \frac{d[P']}{dx} + \frac{dd[Q]}{dx^2} )$ $d\Pi^{iv} = nv.dx ( [L]\alpha + [L']\mathfrak{G} + [L'']\gamma + [L''']\delta + [N'''] - \frac{d[P'']}{dx} + \frac{dd[Q']}{dx^2} )$ $d\Pi^v = nv.dx ( [L]\alpha + [L']\mathfrak{G} + [L'']\gamma + [L''']\delta + [L^{iv}]\epsilon + [N^{iv}] - \frac{d[P''']}{dx} + \frac{dd[Q'']}{dx^2} )$ &c. His comparatis cum valoribus assumtis, erit $\alpha = 0$ $\mathfrak{G} = [L]\alpha dx + \frac{[Q]}{dx}$ $\gamma = dx ( [L]\alpha + [L']\mathfrak{G} + \frac{[P']}{dx} - \frac{[Q]+2d[Q]}{dx^2} )$ $\delta =$