Methodus inveniendi lineas curvas

AD CURVAS INVENIENDAS ABSOLUTA. 107 $\pi' = \int [Z] dx$ $\pi'' = \int [Z] dx + [z] dx$ $\pi''' = \int [Z] dx + [z] dx + [z'] dx$ $\pi'''' = \int [Z] dx + [z] dx + [z'] dx + [z''] dx$ $\pi''''' = \int [Z] dx + [z] dx + [z'] dx + [z''] dx + [z'''] dx$ &c. Ubi notandum est quantitatis $\int [Z] dx$ valorem differentialem esse $=0$, eo quod particula $n\nu$ nullam mutationem infert in abiscissam AL ad quam $\int [Z] dx$ refertur, Tantum igitur terminorum differentialium $[Z]dx$, $[Z']dx$, $[Z'']dx$ &c. valores differentiales investigari oportebit. Erit autem. $d. [Z] dx = dx ( [L] d\pi + \frac{[Q.] n\nu}{dx^2} )$ $d. [Z'] dx = dx ( [L'] d\pi' + \frac{[P'] n\nu}{dx} - \frac{2[Q.'] n\nu}{dx^2} )$ $d. [Z''] dx = dx ( [L''] d\pi'' + [N.''] n\nu - \frac{[P.''] n\nu}{dx} + \frac{[Q.''] n\nu}{dx^2} )$ $d. [Z'''] dx = dx [L'''] d\pi'''$ $d. [Z''''] dx = dx. [L''''] d\pi''''$ &c. Nunc porro definiendi sunt valores differentiales quantitatum $\pi'$, $\pi''$, $\pi'''$, &c. per $n\nu$, quos loco $d\pi$, $d\pi'$, $d\pi''$, &c. substitui oportet. Cum autem sit $\pi = \int [z] dx$, & in $[z]$ differentialia secundum gradum superantia non inesse ponantur, fiet valor differentialis ipsius $\pi$, seu $d\pi=0$, ad sequentium autem quantitatum $\pi'$, $\pi''$, $\pi'''$ &c. valores differentiales inveniendos, notasse conveniet esse $\pi' = \int [z] dx$ $\pi'' = \int [z] dx + [z] dx$ $\pi''' = \int [z] dx + [z] dx + [z'] dx$ $\pi'''' = \int [z] dx + [z] dx + [z'] dx + [z''] dx$ $\pi''''' = \int [z] dx + [z] dx + [z'] dx + [z''] dx + [z'''] dx$ &c. O 2 Erit