Vol 4 Iss. 2

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.

Scientific research carried out jointly by several individuals always gives rise to rumors about the degree of participation of individual researchers in the overall work, and these rumors often take on a highly unpleasant character. To avoid the latter, I consider it necessary to note that in works published jointly on behalf of myself and the names of my students, both the topic and the plan of its elaboration belong to me. Preparatory work, such as preparation of solutions, determination and calculation of their concentration, analysis, etc., is carried out entirely by my employees.

In this study, the conditions formation and certain properties of the following disperse systems were studied: No.1. Disperse systems with a liquid disperse phase xH₂O+ yHCl. No.2. Dispersed systems with a solid disperse phase of CuCl₂ composition. No.3. Disperse system with a solid disperse phase of the composition xH₂O + yHCl·H₂O. No.4. Disperse systems with solid disperse phase CuCl₂ · 2H₂O. No.5. Disperse systems with solid disperse phase CuCl₂·2H₂O and a complex liquid disperse phase of the composition (see the article). No.6. Disperse systems with a liquid disperse phase of the composition (see the article). No.7. Disperse systems with a solid disperse phase of composition (xH₂2O + y.cupric oleate). No.8. Disperse systems with a solid disperse phase of the composition (see the article).

Colloids play such an important role in life (indeed, it can be said without exaggeration that up to 90% of the bodies surrounding us are colloids) that it is not surprising if, even in the most distant epochs, we discover familiarity with processes occurring in colloidal systems. My task is to outline the development of colloidal chemistry and the views currently prevailing in it, and, in view of the abundance of material, it will be necessary to dwell only on the most salient points.

The subject of this brief essay will be methods for producing dispersed phases by means of electrical energy. Electrical methods, which make it possible to obtain dispersoid solutions mainly of elementary, simple substances—metals and metalloids—can be divided into two groups: the first group includes the few cases of atomization observed during cathodic and anodic polarization of certain metals in water or aqueous electrolyte solutions; the second group comprises the most general and practical methods of atomization by means of spark and arc discharge.

The article examines the following issues: obtaining, for any substance, a dispersoid solution of low concentration and of significant concentration, as well as the problems of the crystallizability of colloids, obtaining crystalline substances in the colloidal state; dynamic processes within the dispersion medium as factors of stability, theorems on the stability of dispersoid solutions; dynamic and static chemical compounds, etc.

The article examines the vectorial nature of molecular forces of attraction and clarifies the question of changes in the character and degree of molecular orientation. The author draws the following conclusions: 1. Matter is vectorial in all its aggregate states. 2. By increasing the degree of dispersion of any solid crystalline substance, one can change the degree of its overall orientation; moreover, at extremely high degrees of dispersion, the resulting crystalline systems become practically indistinguishable from liquids in terms of orientation. The work further covers: the gaseous-liquid crystalline state of matter and its universality; the systematics and nomenclature of various kinds of the vectorial state of matter; the fundamental law of dispersoidology and its application.

If we consider only the plane on one side and the homological hyperboloid on the other, we can still recognize the kinship of these linear secunds of points, because the points at infinity of the first system, and consequently their entire linear prima, are homologous to the points of the line of intersection of the hyperboloid with the plane of involution, and consequently to this entire line as the linear prima of extra-elements. From this, in particular, it follows that if three arbitrary points are given in the second system, the spheroprima determined by them is easily obtained as follows: we project these three points through center Z onto the plane, construct a circle through them, and transfer the points of the latter by inverse projection onto the parabolic hyperboloid. The center of this spheroprima is projected in the same way. It is clear that this center on the hyperboloid, with respect to the spheroprima, is the pole of the aforementioned line on the hyperboloid.

Artificial ullmannite NiSbS. A compound corresponding in chemical composition to the mineral ullmannite was obtained by Prof. N. S. Kurnakov and stud. Ya. Posternak (at the St. Petersburg Polytechnic Institute) and was transferred to the Mineralogical Institute of the Mining Institute for crystallographic study. The compound CoSbS. This compound, obtained (also like NiSbS) by student Ya. Posternak, was given to me by Prof. N. S. Kurnakov. The cavity walls in the alloy were dottded with finely tabular crystals, arranged without any regularity, having a strong metallic luster and a steel-gray color.

Ammonium bromostannate (NH₄)₂SnBr₆ was obtained by Prof. N. S. Kurnakov by mixing SnBr₄ + 2(NH₄)Br in an aqueous solution in the presence of hydrobromic acid. Crystals of cubic system, lemon-yellow in color. To determine the structure, about 50 crystals were examined.

When beginning the goniometric study of a crystal, the researcher cannot yet foresee how to appropriately orient the crystal on the goniometer so that, upon completion of the work, the symbol of the complex—the primary goal of any goniometric study—can be derived in the simplest way from the obtained diagram. From this it is clear that, in general, when the question of the correct orientation of a crystallographic complex is resolved, it is finally resolved only after completing a series of measurements, deriving the most significant faces, and calculating the planes—it becomes necessary to transform the projection plane, taking the main zone as the circumference of the projection.

Working in 1902 with concentrated solutions of Mn(CNS)₃ and Ba(CNS)₂, and in 1905-1906 with a series of concentrated solutions of particularly readily soluble salts for the purpose of obtaining jellies of crystalline substances, I noticed that under the influence of these salts, filters swell and become so slimy that they slip through the narrow tube of the funnel in the form of a more or less gelatinous lump. According to my theory of peptization (1907-1908), cellulose is peptized because, at a certain high concentration of salt and a certain high temperature, it must transform into some truly soluble compound. The theory given above (see the article) can be generalized to any dispersoid that hydrolyzes into a soluble compound.

This method is all the more interesting because, compared to other methods of obtaining dispersoid solutions, it possesses many advantages, among which we may note: Its remarkable simplicity. The absence of foreign electrolytes in the solution. The complete definiteness of the systems obtained, from the qualitative and possibly also from the quantitative standpoint, and their purity in this sense.