Vol 5 Iss. 2-3

P. P. von Weymarn kindly sent me a small bottle containing large crystals, along with a letter: "At the bottom of the bottle, where reagents were discarded when washing glassware, crystals of green and yellow color formed; I am sending you these crystals; perhaps they will be of interest for your crystal-chemical analysis." The crystals of different colors turned out to be of different sizes and different shapes. The yellow crystals are distinctly tabular; the thickness of the plates is nearly half a centimeter, and their largest dimension exceeds two centimeters. The greenish crystals are presented in a more isometric form and are at least four times smaller in linear dimensions.

In the spring of 1912, at the invitation of a private individual, I had occasion to carry out some drilling work in the Bukhara domains near the city of Termez, in the lower reaches of the valley of the Surkhan River and the Sherabad-Darya River. During this trip, as well as during some excursions from the city of Termez, I carried out geological observations and collected geological material. In the winter of 1912–1913, this material was examined, and on the basis of the data obtained, the following essay was compiled.

In the note "Polar Relations of Imaginary Triangles and Tetrahedra," we showed that these relations are identical to those defined by a known imaginary ellipse or ellipsoid, with neither passing through the given points. However, from the principles set forth in that note, it follows that definite polar relations of real triangles and tetrahedra may exist. This work constitutes a natural continuation of the preceding note, but pertains to triangles and tetrahedra taken as real.

In the Arctic Ocean, opposite the mouth of the Kolyma River, lies the group of Medvezhyi Islands, numbering six. The closest to the shore and the largest in size is Krestovyi or Krestovskii Island (also known as the First); then come four small islands without names, known only by numbers: the Second, Third, Fourth, and Fifth; and finally, the easternmost is Chetyryokhstolbovoy Island — also known as the Sixth. In the summer of 1912, the Medvezhyi Islands were visited by the Hydrographic Expedition of the Arctic Ocean, consisting of two icebreakers: "Taimyr" and "Vaigach." I, as the geologist of the expedition, managed to land and examine only one of these islands — Chetyryokhstolbovoy.

Having derived a series of principal aggregates, both positional and non-positional, we can now more clearly define the very concept of such aggregates. We call an aggregate principal if it is completely and uniquely derived from a given number of elements, with all these elements playing an equal role in the construction. If we denote these elements by letters and derive the construction of the aggregate, first introducing elements marked by some letters and then by others, then, if the derived aggregate is principal, we can rearrange the letters with respect to the elements differently, and the construction remains valid provided that we retain the original letters in its procedural order.

Yu. V. Wulff kindly drew my attention to a very small but exemplary textbook by Miller, "Tract on Crystallography," published in Cambridge in 1863. This little book is only 86 pages long, but it not only lists and depicts the most important forms in crystallography, but, what is especially characteristic of it, it presents and derives the most important formulas for calculation, and moreover, according to that system original to the author, in which double (anharmonic) ratios predominate. Miller's formulas were the first to introduce the principles of the new geometry into the practice of computational crystallography, although their derivation still relies entirely on the formulas of plane and spherical trigonometry.

The copper ore deposit of the Lipovsky mine (not in operation) resulted from the action of intrusive masses of syenitic magma on strata of sedimentary rocks — limestones. Alongside a minor accumulation of copper and iron ores, PbS and ZnS, a number of typical contact-metamorphic minerals are observed in the contact zone — garnet, augite, diopside, actinolite, and others — and, as a product of pneumatolysis, datolite.

Although the preceding article represents something complete, having carried the principal aggregates up to those determined by six elements, it immediately strikes the eye that the highest of them — hexaseconds — do not represent aggregates of the most general character — conosecunds — but only their special varieties, capable of being generated by straight lines, that is, only linear conosecunds. Only such aggregates are completely and uniquely determined by no more than six points and planes. Since it is precisely conosecunds of a general character that establish a polar correlation between points and planes in space, the proof of the theorem in question reduces to the fact that such a correlation can be established in ∞⁹ ways.

A large number of singular points are known, the positions of which are strictly derived for each given triangle. These points are discovered through the study of various properties of triangles, the number of which is very considerable. Despite all the simplicity of such a figure as an ordinary triangle, its study cannot yet be considered exhausted, and over time, although rarely, new properties are discovered. The problem arises to find the point that serves as the center of the linear prima of rays — the polars of points of the circumscribed circle. To solve this problem, it suffices to find the polars of any two points of the circle with respect to the triangle; the point of their intersection is the desired one. It is precisely this that constitutes the new singular point of the triangle referred to in the title.

In the third issue of Fedorov's "Universal Method," in Chapter VII — "Study of Dispersion" — V. V. Nikitin describes idocrase exhibiting sharp dispersion of birefringence. Furthermore, it possesses the property that within a single grain, in its various sections, the magnitude of birefringence varies. As a result, the interference color is not uniform across the entire extent of the idocrase grains. In sections close to parallel to the optical axis — the fourfold axis of symmetry of the crystal — the colors are arranged in bands parallel to each other: the central part of the grain shows white interference color, followed by a band with a mastix-yellow color, then a purple-red one, and finally, the edges of the grain are colored purple-violet.

As is known, these surfaces of the fourth order can be generated by two projective quadratic primas of planes. The theory of these surfaces is expounded in detailed manuals. In the general case, the surface possesses a hexaprima of points at which two rays of the surface intersect. Planes passing through such a pair of rays can be considered tangent planes, and since the plane section of the surface is in the general case a curve of the fourth order, it is clear that if a plane passes through one of the rays of the surface, this curve decomposes into this ray and a curve of the third order, and if a plane passes through a pair of rays, the same curve decomposes already into this pair and a conoprima; hence we see that tangent planes passing through pairs of rays of the surface intersect this surface further in a conoprima.

In the article on linear aggregates of rays, I showed that the system of rays is not an independent system, but that for it one must adopt a parameter in the form of an extra-ray, necessarily entering into the composition of linear aggregates. Then a linear prima is completely and uniquely determined by two, a linear second by three, and a linear tertia by four arbitrary rays. If three arbitrary rays are given, then together with the constant fourth extra-ray we obtain the necessary and sufficient data for determining the rays of a complete linear secund. Since four arbitrary rays are in the general case intersected by a pair of transversals, real or imaginary, it is clear that we can define a linear second only as the aggregate of rays intersecting a given pair of lines a and b. If, consequently, we intersect the lines a and b with three mutually perpendicular rays, then by these are determined a linear straight tetraprima.

By definition, this surface is generated by two homologous quadratic primas of planes. In the general case, this surface possesses a hexaprima of double points, at each of which two rays of the surface intersect. A correlative transformation yields the same surface, since to each point with two intersecting rays, by which a tangent plane is determined, there corresponds correlatively a tangent plane with two rays in it, intersecting at the point of tangency. Consequently, we can also generate such a surface by a correlative path, that is, determine it by two conoprimas with an established projectivity of points.

Tertias of rays, which represent the so-called null systems and are completely and uniquely determined by five arbitrary rays, are usually called linear in view of the fact that in each plane and from each point of space there exists a prima of rays that intersect at a single point and are contained in a single plane; such a prima of rays in a plane or emanating from a single center is called linear. But for a system of rays determined, as a parameter, by a special extra-ray, such primas are no longer linear, since the latter must necessarily contain this extra-ray. Therefore, the null systems themselves do not represent linear tertias in this system, but are quadratic tertias.

A series of successive improvements in the precise representation of the chemical composition of rocks has shown how to find the figurative point of the chemical composition based on given four ratios of oxides. However, the spatial position of this point is determined not by one, but by two projections onto mutually perpendicular planes, following the method of descriptive geometry. Although the highest achievable simplicity was attained in all other respects, the clarity of the representation still suffered somewhat precisely because of the depiction in two projections. In this article, I aim to systematically present the sequence of all operations necessary for graphical representations so that they become clear even to persons who have no conception of either the tetrahedral scheme or the system of vectorial circles.

As is known from the elementary course in crystallography, to determine the symbol [r₀, r₁, r₂, r₃] of the zone axis of a hexagonally isotropic complex from the symbols of two non-parallel faces (p₀, p₁, p₂, p₃) and (q₀, q₁, q₂, q₃) belonging to this zone, one can use the following method (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).

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).

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.