Consider a differential equation f(x, y, p)= 0, of the first order, but of the degree n, where f is a rational and integral function of (x, y, p) not rationally decomposable into factors: the integral equation contains an arbitrary constant c, and represents therefore a system of curves, for any one of which curves the differential equation is satisfied: the differential equation is assumed to be such that the curves are algebraical curves. The curves in question may be considered as undecomposable curves; in fact, if the curve UαVβWγ … = 0 (composed of the undecomposable curves U = 0, V =0, W =0,‥) satisfies the differential equation, then either the curves U =0, V =0, W =0,‥ each satisfy the differential equation, and instead of the curve UαVβWγ … =0 we have thus the undecomposable curves U =0, V =0, W =0,‥ each satisfying the differential equation; or if any of these curves, for instance W =0, &c., do not satisfy the differential equation, then Wγ, &c. are mere extraneous factors which may and ought to be rejected, and instead of the original curve UαVβWγ… =0, we have the undecomposable curves U =0, V =0 satisfying the differential equation. Assuming, as above, the existence of an algebraical solution, this may always be expressed in the form φ (x, y, c) = 0, where φ is a rational and integral function of (x, y, c), of the degree n as regards the arbitrary constant c: this appears by the consideration that for given values (x0, y0) of (x, y) the differential equation and the integral equation must each of them give the same number of values of p.