Differential equations are considered in relation to theorems related to the Monge and Ampere equations (see 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 of the example remained unfinished.
Дается алгебраическое уравнение F(x,y) = 0, mu степени относительно х и nu степени относительно у. Требуется х и у заменить новыми количествами ɛ и ɳ посредством уравнений. ɛ = j(х,у), ɳ = f(x,y), j и f рациональные функции. За немногими исключениями, которые должны быть в каждом частном случае предметом особого исследования, преобразование (2) будет бирационально. Осуществить это преобразование посредством рациональных действий можно следующими образом. Рассмотрим способ приведения гиперэлиптического интеграла.
We will change x along a closed curve in the positive direction. A full description of the proof is in the article.
Let us take the integral and find a substitution of the lowest degree, through which we achieve the reduction of this integral to an elliptic one.
In order to be useful to the beginners, the author should go into such preliminary details that would be unnecessary in a special mathematical journal (see article). As a result, he obtained a new form of the remainder term.