As is well known in mathematical physics, the question pertaining to this reduces to determining a function ϕ(x, y, t), which depends on the point (x, y) and time t, and satisfies the following conditions (see the article). It is perfectly clear, as M. Vergue also notes in his interesting memoir, that the expansion theorem also holds for waves generated by impulsive forces applied to the surface of the liquid (ondes par impulsion) — the only difference being that sin is replaced by cos in the formula.

To determine whether the remainder Rn of the Taylor series tends to zero as lim n = ∞, it is common practice, as is well known, to represent the remainder in various forms. This is because, for this purpose, one form is often more convenient than another, as can be seen, for example, in the expansion of \( \log(1 + x) \). This method can obviously be generalized to find various forms of the remainder for interpolation formulas as well.

In view of the simplicity of the analytical tools required for the proofs and the absence of any need to study an extensive body of literature on the subject, the problem of finding various forms of the remainder for the Lagrange and Newton interpolation formulas could be of particular interest to those embarking on mathematical research for the first time. It is therefore proposed to these individuals as a topic for independent work, one that is not of a compilatory nature.

In Ritz's method, as is known, the given differential equation is considered as the Euler equation arising from the variation of some integral, wherein the integrand of the latter represents a quadratic form in terms of the unknown function and its first derivative. It therefore seems natural to apply the "closure equation" to the question under consideration, and this will be the subject of the first section of the present article. However, before proceeding to this, it should be noted that the result thus obtained, which is identical to the result of the above-mentioned article, is not without interest, in our opinion, if only for the reason that in this way one obtains, albeit in a particular case, a proof of the convergence of Ritz's process even when the quadratic form under the integral sign to be varied is not a definite positive form.

In various mathematical studies related to the question of the existence of a minimum, it is often useful, as we shall attempt to show below, to apply that fundamental relation in the theory of trigonometric series which was named by Prof. V. A. Steklov the "closure equation" and generalized by him to many other systems of orthogonal functions encountered in analysis and mathematical physics.

T. J. Stieltjes was one of the first mathematicians to address, from a very general point of view, the problem of the convergence of the so-called mechanical quadrature formulas. Recently, a very simple and elegant proof of Stieltjes' result for the case of continuous functions was given by J. V. Uspensky, who employs Weierstrass' fundamental theorem on the approximation of continuous functions by polynomials. However, in view of the importance of the question and some of the consequences arising from it, it seems to us not without interest to treat the problem without the aid of the aforementioned Weierstrass theorem and without preliminary study of a rather complex nature, and this will be the subject of the first section of this work.

The question of the existence of so-called 'fundamental functions' for higher-order differential equations has been the subject of research in a number of works, but undoubtedly further elaboration is possible in the sense of applying various methods to its solution. This article represents an attempt to generalize the method of the American geometer Max Mason, as expounded by him for second-order differential equations, to the case of fourth-order differential equations, to which, as is known, the question of the oscillations of an elastic inhomogeneous rod reduces.

The integrable functions also include those whose discontinuities can be enclosed within intervals, the sum of the lengths of which is arbitrarily small; indeed, by enclosing the points of discontinuity in intervals, we can represent the difference (b - a) as the sum of two parts, of which the first relates to the intervals lying, in turn, within the intervals enclosing the points of discontinuity (see the article). It is enough to show that the difference (b - a) can be made arbitrarily small only under one specific, well-defined law of division, one can then choose such division points that coincide with the endpoints of the intervals enclosing the discontinuities of the function f and then the difference (b - a) will indeed be arbitrarily small. From here we conclude that integrable functions include, inter alia, functions with a finite number of discontinuities, as well as functions whose points of discontinuity, even if infinite in number, have a finite number of so-called limit points.

The presented, highly simple proof, although it uses the lemma of Prof. V. A. Steklov, expressed by formula (41), still differs significantly from the various proofs of the same theorem given by Prof. V. A. Steklov and is closer in its idea to the proof of Prof. Hurwitz (based on the Cesàro-Reyer method of arithmetic means), having, it seems to me, the advantage of greater simplicity, since it is based on the application of a known theorem on term-by-term integration of trigonometric series, which itself represents a special case of the closure theorem.

The solution to one of the fundamental problems of mathematical physics, namely the Dirichlet problem for a sphere, is reduced, as is known, to the question of expanding a so-called “arbitrary” function of two angles into a series arranged according to the spherical Laplace's functions. The possibility of expansion for a function that has two first derivatives has been proven, and by reasoning similar to that presented in our article: “On the theory of trigonometric series”, it can be established that the expansion is also possible for a function that satisfies the Lipchitz’s condition.

When expanding “arbitrary” functions into series using the method of least squares, the coefficients of the series, as is known, are formed according to a precisely defined law. Specifically, the law of formation of the coefficient is the same that would take place in the case of uniform convergence of the series, i.e., in other words, the coefficients acquire the form of the so-called Fourier coefficients and the series will be the so-called Fourier series of the expandable function. (see the article).

In this short essay, devoted primarily to the scholarly and pedagogical work of the late professor, I would first like to share with readers my recollections of the unforgettable, indelible impression which his lectures made on me, as probably they did on almost all of his former students, in their time.