Vol 1 Iss. 2

In order to be useful to the beginners, I must go into such preliminary details that would be superfluous in a special mathematical journal (see the article). As a result, we obtained a novel form of the remainder term.

This note examines the expression of the length of the arcs of an ellipse and a hyperbola using infinite series (see the article). Adding all these equalities and making reductions, we obtain the following formula, which is none other than the Wallis formula. The same result could be obtained by finding the arc length of a hyperbola.

Mining engineer Natsvalov kindly delivered several samples of sublimate crusts formed during roasting of kupfermatte at the Kedabek plant. The specimen was of interest because of its beautifully formed, although mostly in the form of skeletons and growth figures, octahedra with small cube-trancating faces.

This difference is manifested in the movement of the dissolved substance in the layer of solution separating the different faces of the crystal of the substance from which the solution was also obtained. If there were any doubt about the uniformity in this regard between the properties of a natural face and, for example, a plane of cleavage parallel to it, then the mentioned technique provides a means for resolving it.

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.

I intend to present additional theorems concerning linear aggregates in the geometry of vector spheres and quadratic aggregates of ordinary spheres. For a detailed description, as well as a comparison of megaspheres of vector and ordinary spheres and an overview of the system of parallel vectors, please see the article.

First of all, interest was aroused by the huge Baveno twin of orthoclase from the vicinity of Kyakhta, depicted in Fig. 1, and not only due to the clarity of its formation, but also because of the beautiful almandine crystals included in it. A special optical study carried out by V.I. Sokolov showed that the orthoclase substance is intergrown by plagioclase at No. 5, forming well-developed polysynthetic twins according to the albite law, with one individual of the twin being conformably brunted with the orthoclase substance

The so-called colloidal and amorphous states, just like the crystalline one, are universal properties of matter. The colloidal and amorphous states do not exist as separate states of matter but represent one of the forms of crystalline formations. The main reasons for the colloidal and amorphous states are rooted exclusively in the directions that the crystallization process takes, due to the large values of both the relative rate of formation of the substance inside the solution and another factor which depends mainly on the complexity of the substance and the viscosity of the medium in which crystallization occurs. The rates of diffusion and reaction in strong jellies are very low and sometimes almost close to zero. The crystalline structure is determined by the internal symmetry of the individual masses of the substance.

If you have collineations of two systems, then both systems are equal, because both are in the position of involution, and any point a of the system is collinear to the same point a' of the collinear system, regardless of which of these two systems the given point belongs to. But now let’s replace one of the systems with a system similar to it, and take the combined centers of collineation of both systems as the similarity center R. It is clear that under this condition the systems can no longer be brought into the position of involution, and therefore the construction of homologous (collinear) points becomes more complicated, and in any case, for each given point in the set we will obtain two distinct homologous points, depending on which of the systems the given point belongs to.

In brief reports of the 1st issue of vol. I of the "Notes of the Mining Institute" (p. 83), I reported on some experiments on the growth of spheres (hemispheres) prepared from crystals of chrome and potassium alum. Similar experiments were carried out with crystals of rock salt (NaCl) from Stassfurt. A welded piece of such salt was ground into a hemisphere with a diameter of 10 m.m. with a central plane (100) and glued with wax onto a glass hemisphere of the same diameter.

We outline the possibility of a path for the derivation of an indefinite number of new geometric systems. The main feature of the problems of New Geometry is the indefinite multiplicity in the application of theorems, in contrast to the individuality of conditions in the formulation of problems with which ancient and analytical geometry deals. Problems of a metric nature, for this reason, do not fall within the scope of this discipline; however, it would be inaccurate to say that it includes only problems solvable by positional constructions (which is why New Geometry is more often called positional or projective geometry).

In the quartz-epidote tuff schist, from the Akmola region, there are voids filled either with quartz and epidote, or with turbid grains of feldspar. In the latter, epidote sometimes develops, replacing feldspar from the center to the periphery. One of these epidote grains, shown in the attached figure, exhibits zonal structure. The core of the grain, polarizing in 2nd-order yellow and distinctly pleochroic, is bordered by hexagonal zones of lower birefringence (blue and red), almost non-pleochroic; toward the periphery, only one sector remains with colorless layers of low birefringence. Withdifference in birefringence, all these zones simultaneously extinguish and brighten under crossed Nicols, especially clearly when observed in red light, which eliminates dispersion.

Prof. E. Fedorov proposed a method for determining the nature of the dispersion of optic axes using a universal stage and deriving the optical orientation symbol of a mineral. Currently, at the Mineralogical Institute, a series of measurements have been undertaken using this method. This note presents the results of the first (still limited) observations.