Issues of work and pressure during rolling, which are obviously related to one another, have long been the subject of research, both theoretical and experimental. The number of major experimental studies devoted to these issues is small. For my part, I shall dwell only on those conclusions of other researchers whose consideration is prompted by the course of my own investigation (see the article). The theoretical study by Codron was accompanied by laboratory experiments on the rolling of lead bars between rolls of small diameter. I shall discuss these experiments in some detail later, since they, above all, gave me the opportunity to verify the correctness of the theoretical formulas I derived.

To verify to what extent the calculations based on the formulas I derived in previous notes are consistent with the experimental results, I, at the suggestion of Prof. I. A. Time, applied these formulas to the data from experiments carried out on the rolling of sheet iron at the Schulz, Knaudt & Co. plant in Essen. These data are presented in Prof. Time's article: "Indicator Experiments on the Rolling of Steel Rails and Beams" (Gornyi Zhurnal, 1883) and in E. Blass's article: "Zur Theorie des Walzprocesses" (Stahl und Eisen, 1882, No. 7).

In this note, I aim to find the dependence of the pressure per unit area of metal compressed between two parallel planes on the thickness of the compressed piece of metal, all other conditions being equal. The reason underlying the dependence of pressure per unit area on the thickness of the compressed piece of metal is that during compression, accompanied by an increase in the transverse dimensions of the compressed piece, and consequently in the area of contact between the metal and the compressing planes, a frictional force arises between the compressed metal and the compressing planes. Let us now proceed to studying the effect of this force (see the article).

Let us denote by r the radius of the roll and by φ the angle through which the roll has turned during a certain interval of time. We can imagine the rolling process in such a way that the rolled end of the metal bar remains stationary, while the rolls roll along the surface of the bar (see the article).

The question of the shape of a free jet in the discharge of an ideal fluid from a vessel, when the motion takes place in a plane, was first solved by Helmholtz for the case of fluid outflow through a channel projecting into this vessel. Helmholtz's method was then generalized by Kirchhoff, who provided solutions to several problems related to plane fluid motion, including, among others, the problem of fluid outflow through a slit in a plane wall. Both of these problems are discussed by Lamb in his treatise on hydrodynamics, where he applies the method of conformal transformation by Schwarz and Christoffel to derive the equation of the free jet. The purpose of this present note is to apply this same method to derive the equation of the free jet for fluid outflow through an opening in a vessel with plane walls, where the walls converge towards the opening at an angle of 2α.