Vol 5 Iss. 1

The "Yulia" Mine, belonging to the Siberian Copper Joint-Stock Company (formerly the English Yenissei Copper Co., Ltd.), is located 90 versts in a straight line to the northwest of the district town of Minusinsk, in a mountainous area, at an altitude of about 3,000 to 3,500 feet above sea level. The topography of the area consists of a series of individual, rather high hills with relatively gentle slopes. "Yulia" lies almost on the border between the steppe (to the north) and the taiga (to the south). As a general rule, the northern slopes of the hills are covered with dense thickets of larch and birch, while the southern slopes are either bare or have only a grassy cover. The environs of "Yulia" are composed mainly of crystalline, often siliceous, highly metamorphosed limestones with severely disturbed stratification. In many places, the limestones are intruded through and cut by a whole system of individual massifs and dikes of igneous rocks — red-colored syenite porphyries and syenites.

Involutory-collinear transformations belong to the most elementary operations of the new geometry. However, this always refers to the transformation of real geometric figures. The problem of transforming imaginary figures, e.g., imaginary circles, appears not to have been posed and seems incomprehensible. In the study of a system of pairs of rays, this problem presented itself in all its reality in the following form. If two pairs of rays are given, we take them as two pairs of tangents to a parabola, which we easily construct, and thus we find the linear prima of pairs of rays, the centers of which constitute a line, and the pairs of rays themselves are pairs of tangents to the parabola.

One of the very first principles of the new geometry is the theorem according to which projectivity on lines (linear and quadratic) is established by the correspondence of three elements. Therefore, if four arbitrary lines are given on a plane, then each of them, at its intersection with the other three, gives three points, and this is sufficient to establish the projectivity of points on all these lines, because on each of them we have three corresponding points. If spherical aggregates are given partly by real, partly by imaginary conoprimas, then from them it is necessary to construct two linear aggregates of the same degree, for one of which the value of the class of conoprimas must be changed: the real taken as imaginary and vice versa.

The principal classes of hexaprimas, or what are commonly called space curves of the 3rd order, were derived by Seidewitz and are given in the well-known manual by Reye under the names 1) space hyperbola, 2) space ellipse, 3) parabolic hyperbola, and 4) space parabola. This note is the result of the question: is it possible to construct a hexaprima possessing symmetry? The term hexaprima denotes a prima of points that is completely and uniquely determined by six points, and a space curve of the 3rd order is precisely such a curve. We obtain three constructions leading to hexaprimas of three types of symmetry (see the article).

In my previous works, I examined a number of geometric systems whose elements consist of pairs of points. The simplest and most important of these is the system of parallel vectors, then the systems of harmonic segments and vectors, and finally the system of midpoints of harmonic pairs. However, in all of these systems considered, a certain restriction is introduced, either in the form of vectoriality or in the form of a special parameter of the system. Here I intend to consider a system of such elements given without any restriction; that is, I conceive that an element of the system on the plane can be an arbitrary pair of its points, which simultaneously constitutes a segment.

This theorem very simply resolves a problem which can be formulated as follows as a problem of elementary geometry (see the article). Despite all the simplicity of its solution as a problem within the system of circles, it would hardly be solvable on the basis of the theorems of elementary geometry. It is clear that this theorem can be directly transferred to the system of spheres, replacing in it the words "circle" with the words "sphere". For the proof, it suffices to take the center line Q R as the axis of rotation. Thus, in its most general form, the problem of finding the centers of spherothertions of spheres is also solved, a problem which was previously solved by means of formulas.

In previous works, I have developed in detail the system of vectors. But since vectors essentially represent pairs of points, albeit non-equivalent ones (the initial and the terminal), and since rays can be correlated to each point, it is clear that the systems of vectors can be correlative to systems of pairs of rays, which can hardly be called otherwise than vectorial. However, if it is somewhat difficult to reason about vectorial pairs of rays in the plane, then, for the sake of complete clarity, I prefer to establish such a sequence of systems that are included in the composition of all linear primas of this system.

We have come to the conclusion that a system of rays can be introduced into the general series of geometric systems, and we can bring it into a general correlative connection. However, this system no longer belongs to the class of independent systems, but rather to the class of systems restricted by a certain parameter, which we must recognize as some extra-ray, constant for all linear primas, just as we have an analogous system of points with a point as a parameter, included in the composition of all linear primas of that system.

Any aggregate of circles whatsoever will not differ from the collection of circles of the previous system, but will constitute only half of the collection of that system, with the linear primas and seconds of ordinary circles remaining such also for this system; but the linear collections of vectorial circles of the previous system will no longer be such for this system, because the tangent linear primas of the previous system are no longer linear primas of this system. It is easy to prove that in this system aggregates of vectorial circles are altogether absent; indeed, one cannot even postulate them. In fact, if I postulate, for example, a right vectorial circle, then the diametrically opposite one is already a left vectorial circle; we obtain essentially two vectorial circles, by which their linear prima on the sphere is completely and uniquely determined; it is clear that in its presence one cannot postulate a third, arbitrary circle; generally, it would no longer be part of a defined linear prima.

In the article "A Simple and Accurate Representation of Points of Four-Dimensional Space on a Plane by Means of Vectors," not only is the system of vectors on a plane examined in detail, but also the foundations for constructing a linear prima of vectors in space from two given ones are indicated, namely, that this linear prima consists of segments generating a hyperbolic paraboloid, enclosed between two directrices, one of which is the line of initial points and the other the line of terminal points, with the construction itself being achievable by decomposing the two given vectors into components along three coordinate axes and by constructing, from the components, linear primas of parallel vectors; three vectors with a common initial point, but parallel to the coordinate axes, whatever directions we may choose for the latter, are precisely the component vectors of the linear prima; the terminal point of the latter lies on the line of terminal points.

We shall consider the case when a bar with a rectangular cross-section is rolled. 1. We determine, depending on the pressure on the rolls, the pressure per unit area of the metal, assuming this pressure to be uniform. In other words, we determine the average pressure per unit area of the metal (see the article). 2. We calculate the useful mechanical work for a single pass of the bar between the rolls. By useful work we understand here the work that is expended specifically on changing the shape of the bar.

In the summer of 1912, the author of the note visited the middle and lower part of the Kara-Turgai River basin, which occupies the central part of the Turgai district. Along the way, he examined a fairly large number of outcrops, and in some of them a rather rich fauna was encountered. The results of these observations are presented in this note. The central part of the Turgai district of the Turgai region has so far been visited by very few researchers and travelers.

The application of X-rays gave W. L. Bragg (and his father) means which have led to conclusions extremely important for the theory of crystal structure. In part, these conclusions are unexpected, at least in the respect that it was anticipated that the points of regular systems would be seen as the centers of chemical particles, whereas the experiments of the said scientist have led to the conclusion that these are the centers of atoms. Thanks to this, in substances of the simplest chemical composition, special regular systems of points are also obtained, with the centers of symmetry occupied by individual atoms, as if the atoms themselves also possess high symmetry.

Shaitanskaya Dacha is located on the very ridge of the southern part of the Middle Urals, included in Sheet 138 of the general geological map of Russia. To the south, the dacha is bordered by the Chusovaya River, which serves as a natural boundary with Revdinskaya Dacha; to the north and west, it adjoins Bilimbaevskaya Dacha, and to the east, Verkh-Isetskaya Dacha. The western chain consists of individual mountains: Volchikha and the three Magnitnye mountains. The former is composed of gabbro, while the others are composed of amphibolites. The eastern chain, known under the name Grebni (The Ridges), is a purely serpentine range, in the very center of which there are outcrops of primary igneous rocks of ultrabasic magma: dunites and pyroxenites, interconnected by transitional rocks. Let us assign the dunites to the olivine subgroup (see the article).

This note is prompted primarily by the desire to present Bragg's final conclusion in a more graphic form; and then, in view of the complete originality of this conclusion and its rather sharp divergence from our previous ideas about the structure of particles, a desire arose to determine whether it is possible to reconcile it with them. Reflecting on the arrangement of the atoms, we can easily understand that it is of two kinds. Some atoms occupy the positions of the centers of rhombic dodecahedra, others occupy the position of such four trigonal vertices of a dodecahedron that together belong to a tetrahedron. It is precisely this arrangement that determines the hexakis-tetrahedral type of symmetry, and, although the arrangement of the centers of the some particles corresponds to a dodecahedral structure, the situation is altered by the arrangement of the other atoms.

Strictly speaking, as many projections can be constructed as there are geometric systems of the second degree — that is, an unlimited number — and if I now wish to mention such projections as new crystallographic ones, it is solely because they offer peculiar conveniences for solving certain crystallographic problems, conveniences not provided by other projections. Here I have in mind those projections which are obtained from the linear and gnomonic projections if they are subjected to transformation by inverse radii, which is why they can be called grammacyclic and gnomocyclic, respectively.

In previous works, methods for such a determination were given with the aid of the tables of Sokolov and Artemyev, with the exception of the cases listed in the title, unless the pole corresponding to the twofold axis of symmetry is simultaneously the pole of the (1000) face. In all these cases, the performance of specific shears was assumed in order to determine the density of the faces of the principal zone. In this note, I will show that even for these cases one can manage without shear (see the article).

Even from the preceding note, one can see the importance, in the initial study of crystallography, of deriving special elementary formulas for determining the density of isotropic complexes, both cubic and hypohexagonal. Beginners become acquainted with the technique of determining densities from tables best and most easily precisely on examples of isotropic complexes, since the simple formulas provide an ideal check on the determinations made and immediately acquaint them practically with the degree of accuracy—or rather inaccuracy—of graphical operations.

Pascal's theorem lies at the foundation of the theory of conoprimas, expressing their fundamental property of being completely and uniquely determined by five elements. In its modern generalized form, it can be expressed as follows: if six elements ABCDEF of one linear second belong to one conoprim, then, finding the intersection elements of the linear primas AB with DE, BC with EF, and CD with FA, we obtain that all three belong to one linear prima. (see the article). This expression clearly testifies to the deep organic connection of each sixth element with the five remaining ones that determine the conoprima. A simpler analogue of this theorem can be found in the well-known theorems expressing the fundamental properties of spheroprims and spheroseconds.

The known properties of gnomonic projections of trigonaloid crystals prompted me about the presence of the relations mentioned in the title, which seemed to me paradoxical. For the case under consideration, the theory of poles and polars unfolds in its usual form: two points are the poles of two polars and, in turn, determine a line — the polar of the intersection point of these polars. To each vertex of a triangle, the opposite side is polar, etc., and in no case is there a point through which its polar passes, as is the case for imaginary conoprimas of projectivity (see the article).

The content of this note is a direct consequence of the preceding one. It presents a theorem that makes it possible to construct a hexasecond from seven arbitrary points. Since a hexasecond is a positional figure and is correlatively transferred to all geometric systems, the construction of a hexasecond of planes from seven given ones is implied by itself. But if only six planes are given, then the extra-plane is always at our disposal as the seventh and is not counted, being unique in its kind.

In the preface to the collected works of I. P. Dolbnya, published in French by resolution of the Council of the Mining Institute of Empress Catherine II under the title "Oeuvres mathématiques de Jean Dolbnia," the famous French mathematician Gaston Darboux, in particular, wrote: "The distinguished geometer, whose works I have the honor to present to the Academy, was during his lifetime Professor of Analysis at the St. Petersburg Mining Academy, of which he had been a brilliant student; but the majority of his works were published in French in our mathematical collections and principally in the Bulletin of Mathematical Sciences. By deciding to gather them into a single volume and thus present them as a whole, the Mining Academy has rendered a real service to mathematical studies. The works of Dolbnya do honor to his country."