Methodus inveniendi lineas curvas

132 DE USU METHODI I. Maximi minimive formula $\int Zdx$. $dZ = Mdx + Ndy + Pdp + Qdq + Rdr + \&c.$ Valor differentialis erit $\omega y . dx(N - \frac{dP}{dx} + \frac{ddQ}{dx^2} - \frac{d^3 R}{dx^3} + \frac{d^4 S}{dx^4} - \&c.)$ qui valor differentialis pro omni variabilis $x$ magnitudine aeque valet. II. Maximi minimive formula $\int Zdx$. $dZ = Ld\pi + Mdx + Ndy + Pdp + Qdq + \&c.$ $\& \pi = \int [Z] dx$ existente $d[Z] = [M]dx + [N]dy + [P]dp + [Q]dq + [R]dr + \&c.$ Jam posito post integrationem $x=a$, fit $\int Ldx = H$, ponaturque $H - \int Ldx = V$, Valor differentialis erit $\omega y . dx(N + [N]V - \frac{d.(P+[P]V)}{dx} + \frac{dd.(Q+[Q]V)}{dx^2} - \frac{d^3.(R+[R]V)}{dx^3} + \frac{d^4.(S+[S]V)}{dx^4} - \&c.)$ III. Maximi minimive formula $\int Zdx$. $dZ = Ld\pi + Mdx + Ndy + Pdp + Qdq + \&c.$ $\& \pi = \int [Z] dx$ $d[Z] = [L]d\pi + [M]dx + [N]dy + [P]dp + [Q]dq + \&c.$ $\& \varpi = \int [z] dx$ $d[z] = [m]dx + [n]dy + [p]dp + [q]dq + [r]dr + \&c.$ Sit iterum, posito post integrationem $x=a$ ut ante, $\int Ldx = H$