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Date submitted1914-06-18
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Date accepted1914-08-18
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Date published1914-12-01
Polar relations of real triangles and tetrahedrons
- Authors:
- E. S. Fedorov
In the note “Polar relations of imaginary triangles and tetrahedrons” we showed that these relations are identical with those determined by the well-known imaginary ellipse or ellipsoid, and neither one nor the other passes through given points. But from the grounds given in this note , it follows that there can be certain polar relations between real trigons and tetrahedrons. This work is a natural continuation of the previous note, but relates to triangles and tetrahedrons taken to be real.
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Date submitted1914-06-08
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Date accepted1914-08-27
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Date published1914-12-01
Principal aggregates in the systems of points and planes
- Authors:
- E. S. Fedorov
Having derived a number of main aggregates, both positional and non-positional, we can now more clearly define the very concept of such. We call the main set one that can be completely and unambiguously deduced from a given number of elements, if all these elements play the same role in the construction. If we designate these elements by letters and derive the construction of a collection, introducing first the elements marked with certain letters and then with the other ones, then, if the deduced collection is the main one, we can move the letters in relation to the elements differently, and the construction remains valid if in its order during the course of the process, we kept the original letters.
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Date submitted1914-06-18
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Date accepted1914-08-22
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Date published1914-12-01
Supplementing the derivation of major aggregates up to octoprimes and conoseconds
- Authors:
- E. S. Fedorov
Although the previous article represents something complete, bringing the main aggregates to those that are determined by six elements, it is immediately striking that the highest of them - hexaseconds - do not represent aggregates of the most general nature - conoseconds, but only their special differences that can be reproduced by straight lines, that is, only line conoseconds.Only those are completely and unambiguously determined by no more than six points and planes. Since it is conoseconds of a general nature that establish polar correlativity between points and planes in space, the proof of the theorem in question boils down to the fact that such correlativity can be established in ∞⁹ ways
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Date submitted1914-06-05
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Date accepted1914-08-30
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Date published1914-12-01
New special points of triangles in connection with gnomonic projections of crystallographic complexes
- Authors:
- E. S. Fedorov
There is a large number of special points, the position of which is strictly deduced for each given trigon. These points are discovered when studying certain properties of trigons, the number of which is very significant. Despite the simplicity of such an image as an ordinary trigon, its study is still not yet can be considered exhausted and over time, although rarely, new and new properties are discovered. The task is to find the point that serves as the center of the linear primacy of the rays—the polars of the points of the circumscribed circle. To solve this problem, it is enough to find the polars of any two points of the circle in relation to the triangle; the point of their intersection is the desired one. It is this point that constitutes the new special point of the triangle that is mentioned in the title.
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Date submitted1914-06-01
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Date accepted1914-08-24
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Date published1914-12-01
IV order ruled surface with high symmetry and a curve with four cusp points
- Authors:
- E. S. Fedorov
As is known, these fourth-order surfaces can be reproduced by two projective quadratic prima planes. The theory of these surfaces is presented in detailed manuals. In the general case, there is a hexaprima of points on the surface at which two rays of the surface intersect. Planes passing through such a pair of rays can be considered tangent planes, and since a flat section of a surface in the general case is a curve of order IV, it is clear that if a plane passes through one of the rays of the surface, then this curve splits into this ray and a curve of III order, and if the plane passes through a pair of rays, then the same curve breaks up into this pair and another conoprima; from here we see that the tangent planes passing through pairs of rays of the surface intersect this surface while still in conoprima.
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Date submitted1914-06-21
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Date accepted1914-08-27
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Date published1914-12-01
Quadratic ray primes of rays
- Authors:
- E. S. Fedorov
In an article on linear sets of rays, I showed that a system of rays is not an independent system, but that for it it is necessary to take a parameter in the form of an extra ray, which is necessarily part of the linear sets. Then the linear prima is determined completely and unambiguously by two, the linear second by three and the linear third 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 to determine the rays of a full linear second. Since four arbitrary rays in the general case are intersected by a pair of secants, real or imaginary, it is clear that we can define a linear second only as a set of rays intersecting a given pair of straight lines a and b. For simplicity, we will assume that it is real. Let us denote the extraray defined by the extrapoints of these two common secants by e.
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Date submitted1914-06-10
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Date accepted1914-08-03
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Date published1914-12-01
Ruled surface of the VI order as a hexaprima of rays
- Authors:
- E. S. Fedorov
By the definition of this surface, it is reproduced by two homologous quadratic primes of planes. In the general case, on this surface there is a hexaprima of double points, at each of which two rays of the surface intersect. The correlative transformation gives the same surface, since each point with two intersecting rays, which define a tangent plane, has a correlative tangent plane with two rays in it intersecting at the point of tangency. Consequently, we can reproduce such a surface in a correlative way, that is, we can define it by two conoprimes with established projectivity of points.
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Date submitted1914-06-16
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Date accepted1914-08-24
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Date published1914-12-01
Quadratic thirds and seconds of rays
- Authors:
- E. S. Fedorov
The thirds of rays, representing the so-called zero systems and 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 in space there is a prima of rays intersecting at one point and contained in one plane; such a prima of rays in a plane or emanating from one center is called linear. But for a system of rays, defined as a parameter by a special extra-ray, such primes are no longer linear, since the latter must necessarily contain this extra-ray. Therefore, the zero systems themselves do not represent linear thirds in this system, but these are quadratic thirds.
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Date submitted1914-06-21
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Date accepted1914-08-19
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Date published1914-12-01
A visual representation of the chemical composition of rocks from the Christiania region and lavas of the Caucasus
- Authors:
- E. S. Fedorov
Ряд последовательных усовершенствований в точном изображении химического состава горных пород показал, как найти фигуративную точку химического состава по данным четырем отношениям окислов. Однако пространственное положение этой точки определяется не одною, а двумя проекциями на взаимноперпендикулярных плоскостях по методу начертательной геометрии. Хотя во всех других отношениях была достигнута высшая достижимая простота, но все-таки наглядность в изображении несколько страдала именно вследствие изображения в двух проекциях. В этой статье я имею в виду систематически изложить ход всех операций, необходимых для графических изображений так, чтобы они стали ясны даже для лиц, не имеющих понятия ни о тетраэдрической схеме, ни о системе векторальных кругов.
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Date submitted1914-06-15
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Date accepted1914-08-25
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Date published1914-12-01
Relationships between face and edge symbols in hypohexagonal crystals
- Authors:
- D. N. Artem'ev
As is known from the elementary course of crystallography, to determine the symbol [r0, r1, r2, r3] of the belt axis of a hexagonal-isotropic complex from the symbols of two non-parallel faces (p0, p1, p2, p3) and (q0, q1, q2, q3) belonging to this belt, you can use the following technique (see article).
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Date submitted1914-06-10
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Date accepted1914-08-28
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Date published1914-12-01
Resistance of metal to compression between two rolls during rolling
- Authors:
- S. N. Petrov
Let us denote by r the radius of the roll and by φ the angle through which the roll has turned over a certain period of time. We can imagine the rolling process in such a way that the rolled end of a metal bar remains motionless, and the rollers roll along the surface of the bar (see article).
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Date submitted1914-06-26
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Date accepted1914-08-13
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Date published1914-12-01
Resistance of a tensile metal to compression between two parallel planes
- Authors:
- S. N. Petrov
In this note, I aim to find the dependence of the pressure per unit surface of a metal compressed between two parallel planes on the thickness of the compressed piece of metal, considering other conditions equal. The reason that determines the dependence of the pressure per unit surface on the thickness of the compressed piece of metal is that during compression the associated increase in the transverse dimensions of the compressed piece, and therefore the area of contact of the metal with the compressive planes, a friction force arises between the compressed metal and the compressive planes . Let us now study the influence of this force (see article).
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Date submitted1914-06-12
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Date accepted1914-08-06
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Date published1914-12-01
On the question of the shape of a free jet when a perfect liquid flows out of a vessel with flat walls
- Authors:
- S. N. Petrov
The question of the form of a free jet when a perfect liquid flows out of a vessel, when the movement occurs in a plane, was first solved by Helmholtz for the case of liquid flow through a channel protruding into this vessel. Helmholtz’s method was then generalized by Kirchhoff, who gave a solution to several problems related to plane the movement of a fluid, and, by the way, the problem of the flow of fluid through a slot in a flat wall.Both of these problems are considered by Lamb in his treatise on hydrodynamics, and he uses the method of conformal transformations of Schwartz and Christoffel to derive the equation of a free jet. The purpose of my present note is to apply the same method to the derivation of the equation of a free jet when a liquid flows through a hole in a vessel with flat walls, and the walls converge to the hole at an angle of 2a.
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Date submitted1914-06-29
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Date accepted1914-08-10
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Date published1914-12-01
Geological research in the Central Bukhara
- Authors:
- S. M. Mikhailovskii
In the spring of 1912, at the invitation of a private person, I had to carry out some drilling work in the Bukhara area near the city of Termez in the lower reaches of the valley of the Surkhana and Shirabad-Darya rivers. 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 reviewed and, based on the data obtained, the following essay was compiled.
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Date submitted1914-06-25
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Date accepted1914-08-24
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Date published1914-12-01
About the geology of Chetyrehstolbovyi Island
- Authors:
- N. Kirichenko
In the Arctic Ocean, opposite the mouth of the Kolyma River, there is a group of Medvezh'i Islands numbering six. The closest to the shore and the largest in size is the island of Krestovyi or Krestovskii (also the first); then there are four small islands that have no name, known only under the numbers: second, third, fourth and fifth; and finally, the easternmost is the Chetyrekgstolbovyi Island - also the sixth. In the summer of 1912, the Medvezh'i Islands were visited by the Hydrographic Expedition of the Arctic Ocean, consisting of two icebreakers: Taimyr and Vaygach. I, as the geologist of the expedition, managed to land and inspect only one of these islands - Chetyrekhstolbovyi.
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Date submitted1914-06-06
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Date accepted1914-08-30
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Date published1914-12-01
Datolite from Lipovsky copper mine, Nizhnii. Tagil district
- Authors:
- N. Yu. Fedorov
The copper ore deposit of the Lipovskii mine (not in operation) was the result of the influence of intrusive masses of syenite magma on the strata of sedimentary rocks—limestones. Along with an insignificant accumulation of copper and iron ores, PbS and ZnS, in the contact area there is a number of typical contact-metamorphic minerals—garnet, augite, diopside, actinolite, etc., and, as a product of pneumatolysis, datolite.
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Date submitted1914-06-27
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Date accepted1914-08-25
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Date published1914-12-01
About the dispersion of idocrase
- Authors:
- I. Ansheles
In the third edition of “Fedorov’s Universal Method,” in Chapter VII — “dispersion research” — V.V. Nikitin describes idocrase, which has a sharp birefringence dispersion. In addition, it also has the property that within the same grain, in different parts of it, the magnitude of birefringence is different. Due to this, the interference coloring is not uniform throughout the idocrase grains. In sections close to parallelism to the optical axis — the quadruple axis of symmetry of the crystal — the colors are arranged in stripes parallel to each other: the central part of the grain has a white interference color, followed by a stripe with a mastic yellow color, then with a purple-red color, and finally, the edges of the grain are colored in a purple-violet color.
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Date submitted1914-06-09
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Date accepted1914-08-19
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Date published1914-12-01
Analysis of crystals released from laboratory wastewater
- Authors:
- E. S. Fedorov
P.P.f. Weymar kindly sent me a bottle with large crystals in a letter: “At the bottom of the bottle, where the reagents were poured out when washing dishes, green and yellow crystals formed; I am sending these crystals to you; perhaps they are interesting for your crystallo- chemical analysis". Crystals of different colors turned out to be of different sizes and different appearances. The yellow crystals are clearly lamellar; The thickness of the plates is almost half a centimeter, and their largest size exceeds two centimeters. The greenish crystals are presented in a more isometric form and are at least four times smaller in linear dimensions.
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Date submitted1914-06-09
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Date accepted1914-08-13
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Date published1914-12-01
Miller's important formula
- Authors:
- E. S. Fedorov
Yu. V. Wolf kindly drew my attention to Miller’s very small but exemplary textbook "Tract on crystallography", published in Cambridge in 1863. This is a book of only 86 pages, but it not only lists and depicts the most important forms of a crystallographer, but, what is especially characteristic of it, the most important formulas for calculations are presented and derived, and, moreover, according to the system original to the author, in which double (anharmonic) relations predominate . Miller's formulas were the first to introduce the beginning of a new geometry into the practice of computational crystallography, although their derivation is still entirely based on the formulas of plane and spherical trigonometry.