Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 79 tur, habebiturque $0 = A - P + \frac{dQ}{dx} - \frac{ddR}{dx^2} + \frac{d^3S}{dx^3} - \&c.$ quae multiplicetur per $dp = qdx$, ad eamque addatur $0 = dZ + Pdp + Qdq + Rdr + Sds + \&c.$ quo facto prodibit ista aequatio integrabilis $0 = Adp - dZ + qdQ - \frac{qddR}{dx} + \frac{q d^3 S}{dx^2} - \&c. + Qdq + Rdr + Sds + \&c.$ cujus integrale erit $0 = Ap - B - Z + Qq - \frac{qdR}{dx} + \frac{qddS}{dx^2} \&c. + Rr - \frac{rdS}{dx} + Ss$ seu $0 = Ap - B - Z + Qq - \frac{qdR - Rdq}{dx} + \frac{qddS - dqdS + Sddq}{dx^2} - \frac{q d^3 T - dqddT + d T ddq - T d^3 q}{dx^3} \&c.$ CASUS VI. §. 68. Sit $M=0$ & $N=0$ & $P=0$, ita ut fit $dZ = Qdq + Rdr + Sds + Tdt + \&c.$ Ob $N=0$ & $P=0$, per Casum secundum, duae integrationes locum habent, eritque aequatio, pro curva quaesita, haec, $0 = Ax - B + Q - \frac{dR}{dx} + \frac{ddS}{dx^2} - \frac{d^3T}{dx^3} + \&c.$ ad quam per $dq = rdx$ multiplicatam si addatur $0 = dZ - Qdq - Rdr - Sds - Tdt - \&c.$ habebitur ista aequatio denuo integrabilis: $0 = Axdq - Bdq + dZ - r dR + \frac{rddS}{dx} - \frac{rd^3T}{dx^2} + \&c. - Rdr - Sds - Tdt - \&c.$ cujus integrale est $0 =$