Methodus inveniendi lineas curvas

98 DE METHODO MAX. ET MIN. $d.Zdx = dx(Ldn + Mdx + Ndy + Pdp + \&c.)$ $d.Z'dx = dx(L'dn' + M'dx + N'dy' + P'dp' + \&c.)$ $d.Z"dx = dx(L"dn" + M"dx + N"dy" + P"dp" + \&c.)$ $\&c.$ Quod si nunc loco differentialium $dn, dn', dn"$ \&c. $dy, dy', dy"$, \&c. $dp, dp', dp"$, \&c. $dq, dq', dq"$, \&c. valores supra inventi substituantur, & eodem modo, quo ante usi sumus, in unam summam conferantur, prodibit formulae $\int Zdx$ pro abscissa $AZ = a$ valor differentialis = $= nv.dx \left( [N](H-\int Ldx) - \frac{d.[P](H-\int Ldx)}{dx} + \frac{dd.[Q](H-\int Ldx)}{dx^2} \right.$ $- \frac{d^3.[R](H-\int Ldx)}{dx^3} + \frac{d^4.[S](H-\int Ldx)}{dx^4} - \&c. )$ $+ nv. dx \left( N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \&c. \right).$ Atque ex hoc resultabit aequatio pro curva quaesita haec : $0 = [N](H-\int Ldx) - \frac{d.[P](H-\int Ldx)}{dx}$ $+ \frac{dd.[Q](H-\int Ldx)}{dx^2} - \frac{d^3.[R](H-\int Ldx)}{dx^3} + \&c.$ $+ N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \&c.$ ubi notandum esse $H$ valorem formulae $\int Ldx$, qui oritur posito $x=a$. Q. E. I. COROLL. I. 20. Regula igitur Capite praecedente inventa amplior est reddita; nunc enim curvam definire possumus, maximum minimumve habentem valorem formulae $\int Zdx$, si $Z$ non solum est functio quantitatum determinatarum $x, y, p, q, r, \&c.$ sed etiam unam quantitatem integralem indefinitam $\int [Z]dx$ in se complectitur: dummodo $[Z]$ fit functio determinata. Co-