Differential equations are considered with respect to theorems related to the Monge and Ampère equations (see the article).

We have obtained a homogeneous linear differential equation of the first order with partial derivatives with respect to q. This equation is equivalent to a system of ordinary cumulative differential equations of the first order (see article). This note was found in the postmortem papers of I. P. Dolbnya. Only the calculations for the example remained unfinished.

An algebraic equation F(x,y) = 0 is given, of degree μ with respect to x and degree ν with respect to y. We are required to replace x and y by new quantities ε and η by means of the equations. It is required to replace x and y with new quantities ɛ and ɳ by means of equations. ɛ = j(x,y), ɳ = f(x,y), j and f are rational functions. With a few exceptions, which should be the subject of special research in each particular case, transformation (2) will be birational. This transformation can be carried out through rational actions in the following ways. Let's consider a way to reduce a hyperelliptic integral.

Let us vary x along a closed curve in the positive direction. The full description of the proof is provided in the article.

Let us take the integral and find a substitution of the lowest degree, by means of which we can achieve the reduction of this integral to an elliptic one.

In order to be useful to the beginners, I must go into such preliminary details that would be superfluous in a special mathematical journal (see the article). As a result, we obtained a novel form of the remainder term.