This question for the case of a homogeneous field—the field of gravity—was discussed in detail by Prof. M.I. Akimov, who found all surfaces that permit the motion of apoint mass under gravity along helical lines with a vertical axis in the presence of friction, and investigated the conditions for these surfaces under which the motion in question would be stable.

Let FT and Fn be the absolute values of the tangential and normal to the surface components of the weight force acting on a point. A point placed on the surface without initial velocity remains in equilibrium in the presence of friction if FT≤kFn, and begins to move if FT>kFn, where k is the coefficient of friction. The differential equations of motion of a point M of weight g on a rough surface (r, φ, z)=0 in cylindrical coordinates r, φ, z (the z-axis is directed vertically downward) are given in the article.

The article contains the calculation of the tables necessary for the construction of helical surfaces z=aφ+f(r)z=aφ+f(r), which are encountered in the theory of the spiral separator. (See the work of M. I. Akimoff, "Motion of a heavy point on a helix located on a frosted surface" in the Annals of the Mining Institute in Leningrad, Vol. 10, Issue 1 (1936), pp. 11 and 21, Vol. 12, Issue 3 (1939), pp. 4, 5, 38 and 39. See also the article by V. A. Egonnoff, "On the theory of the spiral separator" in the same scientific publications, Vol. 10, Issue 1, pp. 23–30.)

Generalizing the Catalan problem about the motion of a point rotating on a polished surface, the author searches for surfaces that allow the motion of a point rotating in a spiral with a vertical axis in the presence of friction, and explores the nature of the stability of the motion of a rotating point on a spiral located on a rough helical surface with a vertical axis, as well as the influence on this motion of the resistance of the medium, depending on speed, and the rotation of the surface around aerodynamic propellers.

In this work, which presents a detailed and systematic exposition of the results I obtained earlier,elementary derivations are presented, and a number of new surfaces are indicated that may find application in the theory of the spiral separator and which, with an appropriate choice, offer certain advantages over the ordinary inclined helical surface—both in terms of achieving greater stability of the helical motion occurring on them, and in terms of the possibility of suitably selecting the velocity distribution of the masses moving along them. This work also presents new results concerning the investigation of the stability of the considered motion along a helical line on a rough surface, and clarifies the influence on this motion of: (1) the resistance of the medium, (2) the rotation of the surface about the vertical axis of the helical lines lying on it.

The purpose of this note is to simplify the method proposed by Academician N. M. Krylov for expanding a given function in a series in Jacobi polynomials. By constructing a priori an expansion with uniform and absolute convergence, we show, without relying on the Riesz-Fischer theorem, that it can be identified with expansions in Jacobi polynomials.

Chapter IV. Some series expansions in generalized Bessel functions. Chapter V. Application of generalized Bessel functions to problems of mechanics.

The subject of the present work is the study of functions which initially appear in the form of a definite integral (see the article). In the special case of a single variable, we have the classical Kepler equation and its solution given by Bessel. As a simplest example illustrating the application of the obtained formulas, I present the classical problem of the motion of a spherical pendulum in the case of its small oscillations about the lowest equilibrium position. As examples of the application of generalized Bessel functions, I also present the equation of the mathematical pendulum, the equation of the theory of longitudinal bending, the differential equation encountered in the theory of small oscillations in a vacuum of a system with one degree of freedom, as well as in many recent investigations in celestial mechanics, and the differential equation of small oscillations in a resisting medium when the resistance of the medium is proportional to the square of the velocity.

A detailed description of the first proof of Gauss's fundamental theorem and of Cauchy's proof is given (see the article).

Let us investigate the limiting cases of the Riemann function. Let us take the differential equation of the P function and examine the functions (see the article). Following Riemann's example, we imagine the path of integration as a flexible, stretchable and easily movable thread. In its motion, a special point deforms this path, pushing it ahead and never crossing it. With this representation of the integration path, from closed curves corresponding to integrals (11), (13), (15), for integrals (16), (18), (20) we obtain open paths in certain directions extending to infinity.