Chapter 1 contains some historical remarks. The importance of the work of Charpit is convincingly pointed out, and the double name of the classical method of LagrangeCharpit is justified. In Cha pter 2 classical methods for integration of linear systems of partial equations of the first order are briefly described. In Chapter 3 the notion of a regular integral element is introduced and some of its applications indicated. By a regular integral element of a partial equation \( F\left(x_{1}, \ldots, x_{n} ; p_{1}, p_{2}, \ldots, p_{n}\right)=0 \) the author means a set of \( (n-1) \) equations \( F_{n}(x, p)=C_{n}(n=1,2, \ldots, n-1) \) such that the system \( F_{1}=C_{1}, \ldots, F_{n-1}=C_{n-1}, F=0 \) can be solved with respect to \( p_{1}, \ldots, p_{n} \), giving \( n \) functions of the \( x^{\prime} \)’s and \( C^{\prime} \) s, that are partial derivatives of the same function \( V\left(x_{1}, \ldots, x_{n} ; C_{1}, \ldots, C_{n-1}\right) \). In Chapter 4 canonical and Jacobian systems and some of their generalizations are discussed. A brief sketch of Cauchy-Jacobi’s method of characteristics is given. A generalization of the notion of the integral element is discussed in Chapter 5, according to the ideas of Liouville, S. Lie and the author, in the case of a single partial equation. In Chapter 6 these ideas are extended to the case of systems of partial equations of the first order. The bibliography appended at the end of the monograph obviously does not claim to be complete. Some names should have been included there, however, such as the name of Steklov, for instance. Author’s own investigation occupy prominent place in the monograph, but some important classical theories are excluded from the scope of it, such as the theory of contact transformations (about \( 1 / 3 \) of a page at the end of the book).

Bericht über die klassischen Lösungsmethoden der partiellen Differentialgleichungen erster Ordnung mit einer unbekannten Funktion in ihrer geschichtlichen Entwicklung samt den Ergänzungen, die vom Verf. herrühren (1925; F. d. M. 51, 354 (JFM 51.0354.*)).