Methodus inveniendi lineas curvas

IN RESOLVENDIS QUAESTIONIBUS. 133 $=H$, ac ponatur $H - \int L dx = V$. Jam integretur $\int [L] V dx$, fitque integrale, eo casu quo $x=a$ ponitur, $=G$, ac ponatur $G - \int [L] V dx = [V] = G - \int [L] dx (H - \int L dx)$; His positis, Valor differentialis erit $n.dx(N+[N]V+[n][V] - \frac{d.(P+[P]V+[p][V])}{dx} + \frac{d^2.(Q+[Q]V+[q][V])}{dx^2} - \frac{d^3.(R+[R]V+[r][V])}{dx^3} + \frac{d^4.(S+[S]V+[s][V])}{dx^4} - \&c.)$ unde simul lex progressionis patet, si adhuc plura integralia involvantur. I V. Maximi minimive formula $\int Z dx$. $dZ=Ldn+Mdx+Ndy+Pdp+Qdq+\&c.$ $\& n=\int Zdx$ Abeat, posito $x=a$, haec expressio $e^{\int L dx}$ in $H$, denotante $e$ numerum cujus logarithmus est $=1$, fitque $H e^{-\int L dx}=V$: Valor differentialis erit $n.dx(NV - \frac{d.PV}{dx} + \frac{d^2.QV}{dx^2} - \frac{d^3.RV}{dx^3} + \frac{d^4.SV}{dx^4} - \&c.)$ V. Maximi minimive formula $\int Z dx$. $dz=Ldn+Mdx+Ndy+Pdp+Qdq+Rdr+\&c.$ $\& n=\int [Z]dx$ $d[Z]=[L]dn+[M]dx+[N]dy+[P]dp+[Q]dq+[R]dr+\&c.$ Sit, si ponatur $x=a$, post integrationem $\int e^{\int [L]dx} Ldx = H$ atque ponatur R 3