Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 89 $d. Zdx = Ldx. dn$ $d. Z'dx = L'dx. dn'$ $d. Z''dx = L''dx. dn''$ $d. Z'''dx = L'''dx. dn'''$ $d. Z''''dx = L''''dx. dn''''$ Quodsi jam loco differentialium $dn, dn', dn'', dn''', \&c.$ valores supra inventos ex translatione puncti $n$ in $v$ ortos substituamus obtinebimus. $d. Zdx = 0.$ $d. Z'dx = nv. L'dx^2. \frac{[T]}{dx^2}$ $d. Z''dx = nv. L''dx^2 (\frac{[S']}{dx^4} - \frac{4[T]+\int d[T]}{dx^5})$ $d. Z'''dx = nv. L'''dx^2 (\frac{[R'']}{dx^6} - \frac{3[S']+4\int d[S']}{dx^7} + \frac{6[T]+15\int d[T]+10dd[T]}{dx^8})$ $d. Z''''dx = nv. L''''dx^2 (\frac{[Q''']}{dx^8} - \frac{2[R'']+3d[R'']}{dx^9} + \frac{3[S']+8d[S']+6dd[S']}{dx^{10}} - \frac{4[T]+15d[T]+20dd[T]+10d^3[T]}{dx^{11}})$ $d. Z^V dx = nv. L^V dx^2 (\frac{[P'''']}{dx^{10}} - \frac{[Q''']+2d[Q''']}{dx^{11}} + \frac{[R'']+3d[R'']+3dd[R'']}{dx^{12}} - \frac{[S']+4d[S']+6dd[S']+4d^3[S']}{dx^{13}} + \frac{[T]+5d[T]+10dd[T]+10d^3[T]+5d^4[T]}{dx^{14}})$ $d. Z^{VI} dx = nv. L^{VI} dx^2 ([N^V] - \frac{d[P'''']}{dx} + \frac{dd[Q''']}{dx^2} - \frac{d^3[R'']}{dx^3} + \frac{d^4[S']}{dx^4} - \frac{d^5[T]}{dx^5})$ $d. Z^{VII} dx = nv. L^{VII} dx^2 ([N^{VI}] - \frac{d[P^V]}{dx} + \frac{dd[Q'''']}{dx^2} - \frac{d^3[R''']}{dx^3} + \frac{d^4[S'']}{dx^4} - \frac{d^5[T']}{dx^5})$ &c. Sequentium scilicet terminorum incrementa eadem hac lege progrediuntur. Addantur jam senorum priorum terminorum incrementa, prodibit terminorum $Zdx + Z'dx + Z''dx + Z'''dx + Z''''dx + Z^Vdx$ incrementum totale = $nv.dx^2 (\frac{L''[P'''']}{dx^{10}} - \frac{[Q''']dL''+2L''d[Q''']}{dx^{11}} + \frac{[R'']ddL'''+3d[R'']dL'''+3L'''dd[R'']}{dx^{12}} - \frac{[S']d^3L''''+4d[S']ddL''''+6dd[S']dL''''+4L''''d^3[S']}{dx^{13}} \dots\dots\dots +$ Euleri de Max. & Min. M +