Methodus inveniendi lineas curvas

74 DE METHODO MAX. ET MIN. $-P^{IV} + P''' = -dP'''$ $+Q^V - 2Q^{IV} + Q''' = +ddQ'''$ $-R^{VI} + 3R^V - 3R^{IV} + R''' = -d^3 R'''$ $+S^{VII} - 4S^{VI} + 6S^V - 4S^{IV} + S''' = +d^4 S'''$ $-T^{VIII} + 5T^{VII} - 10T^{VI} + 10T^V - 5T^{IV} + T''' = -d^5 T'''$ Quamobrem formulæ $\int Z dx$ integer valor differentialis ex particula $n\nu$ ortus, erit $= n\nu.dx(N^V - \frac{dP^{IV}}{dx} + \frac{ddQ'''}{dx^2} - \frac{d^3R''}{dx^3} + \frac{d^4S'}{dx^4} - \frac{d^5T}{dx^5})$. Hic autem, quia omnes termini sunt homogenei, signaturæ tuto omitti possunt, evanescit enim discrimen inter $N^V$ & $N$, itemque inter $dP^{IV}$ & $dP$, reliquaque. Quocirca habebitur formulæ $\int Z dx$ iste valor differentialis $n\nu.dx(N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \frac{d^5T}{dx^5}):$ ex quo simul valor differentialis formulæ $\int Zdx$ colligi potest; si in $Z$ altiora etiam differentialia inessent. Quare si curva quæratur, quæ habeat $\int Zdx$ maximum vel minimum pro data abscissa, fueritque $dZ = Mdx + Ndy + Pdp + Qdq + Rdr + Sds + Tdt + \&c.$ erit primum formulæ $\int Zdx$ valor differentialis hic: $n\nu.dx(N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \frac{d^5T}{dx^5} + \&c.)$ Hincque pro curva quæsita orietur ista æquatio $0 = N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3R}{dx^3} + \frac{d^4S}{dx^4} - \frac{d^5T}{dx^5} + \&c.$ Q. E. I. COROLL. I. 57. In formula $\int Zdx$, uti eam tractavimus, quantitas $Z$ continet differentialia quinti gradus: si quidem in differentiali ipsius