Vol 6 Iss. 1

In the issue of the journal "The Mineralogical Magazine" (No. 81, September 1915), a note by G. Césaro is published, in which the author derives a formula concerning the numerical property of the set of axes of symmetry located in the planes of symmetry of a polyhedron. From the very title of the article, it is evident that the author was interested not in the application of the formula for determining planes of symmetry—which is quite limited, as it applies only to cases where the planes of symmetry pass through all axes of symmetry (i.e., only to mirror types of symmetry, when symmetric figures can be reproduced in zonohedral mirrors)—but rather in the formula itself with its numerical relationships. However, the author was completely unaware that this very formula had been derived by the author [i.e., the writer of the present text] for a very wide application to zonohedra (and in this sense was called zonohedral) and to paired polyhedra (in which case it is called polar zonohedral).

As is well known in mathematical physics, the question pertaining to this reduces to determining a function ϕ(x, y, t), which depends on the point (x, y) and time t, and satisfies the following conditions (see the article). It is perfectly clear, as M. Vergue also notes in his interesting memoir, that the expansion theorem also holds for waves generated by impulsive forces applied to the surface of the liquid (ondes par impulsion) — the only difference being that sin is replaced by cos in the formula.

The author dwells on the issue of the differentiation of the primary planetary magma (see the article). Besides the factors that operated and determined magmatic differentiation during the period of the planet's cooling and solidification, other factors subsequently came into play, continuing the work of their predecessors. These factors must be recognized as water and the atmosphere after their appearance on Earth. These agents continued to differentiate the rocks, separating bases and acids and concentrating them in individual locations, for example, depositing thick strata of limestones, quartz sandstones, alkali chlorides, and other compounds. These concentrated bases and acids, when reabsorbed by the magma, were possibly the cause of the disruption of homogeneity and the excessive enrichment in bases, which gave rise to the appearance of basic rocks, both alkaline and alkaline-earth. Adopting such a viewpoint, we can, starting from a single primary magma, for example, a granitic magma, explain all the diversity of magmas that have given rise to the magmatic rocks exposed on the Earth's surface, with which we are gradually becoming acquainted.

To determine whether the remainder Rn of the Taylor series tends to zero as lim n = ∞, it is common practice, as is well known, to represent the remainder in various forms. This is because, for this purpose, one form is often more convenient than another, as can be seen, for example, in the expansion of \( \log(1 + x) \). This method can obviously be generalized to find various forms of the remainder for interpolation formulas as well.

The goal of these preparations is to enable, in all cases where laboratory crystals are obtained from supersaturated solutions, an optical study to be carried out in parallel with goniometric measurement, with the most precise possible orientation of the axes of the optical ellipsoid. Until recent years, I used homemade preparations, assembling them from offcuts of cover glasses, cut and glued onto a slide in such a way as to create the required wedge-shaped space for crystallization. I have now settled on a relatively simple and, for research purposes, very convenient type, which is fully clarified by the attached plan and section of the microscopic preparation (see the article).

In view of the simplicity of the analytical tools required for the proofs and the absence of any need to study an extensive body of literature on the subject, the problem of finding various forms of the remainder for the Lagrange and Newton interpolation formulas could be of particular interest to those embarking on mathematical research for the first time. It is therefore proposed to these individuals as a topic for independent work, one that is not of a compilatory nature.

I mean to confine myself only to the simpler conclusions that I use in my lectures (see the article). From this triplicity we deduce the necessity of limitations for the principal numbers of the symbol of a complex, namely: for an octahedral structure, one cannot accept a number greater than 63 1/2°; for a hexahedral structure, only numbers within the range of 45° to 63 1/2° can be accepted; and for a dodecahedral structure, only numbers greater than 45°.

These few examples (see the article) provide highly important indications of the dependence of crystallization on the arrangement of atoms in particles of complex structure. In such particles, it is therefore necessary to distinguish between the central and peripheral parts (in simple particles, e.g., ClNa, this is impossible), and only the latter, by the density of the arrangement of atoms, determines the position of possible faces.

In Ritz's method, as is known, the given differential equation is considered as the Euler equation arising from the variation of some integral, wherein the integrand of the latter represents a quadratic form in terms of the unknown function and its first derivative. It therefore seems natural to apply the "closure equation" to the question under consideration, and this will be the subject of the first section of the present article. However, before proceeding to this, it should be noted that the result thus obtained, which is identical to the result of the above-mentioned article, is not without interest, in our opinion, if only for the reason that in this way one obtains, albeit in a particular case, a proof of the convergence of Ritz's process even when the quadratic form under the integral sign to be varied is not a definite positive form.

According to the essence of the fundamental law of crystal chemistry, the position of each individual atom is expressed in coordinates (see the article), where both the numerators and the denominators are integers, and moreover, the denominators are necessarily numbers greater than the numerators if the points are located inside the elementary parallelohedron or on its surface. Using the results, we can easily and directly obtain, from the symbols of three points, the symbol of the face passing through them.

In various mathematical studies related to the question of the existence of a minimum, it is often useful, as we shall attempt to show below, to apply that fundamental relation in the theory of trigonometric series which was named by Prof. V. A. Steklov the "closure equation" and generalized by him to many other systems of orthogonal functions encountered in analysis and mathematical physics.

To verify to what extent the calculations based on the formulas I derived in previous notes are consistent with the experimental results, I, at the suggestion of Prof. I. A. Time, applied these formulas to the data from experiments carried out on the rolling of sheet iron at the Schulz, Knaudt & Co. plant in Essen. These data are presented in Prof. Time's article: "Indicator Experiments on the Rolling of Steel Rails and Beams" (Gornyi Zhurnal, 1883) and in E. Blass's article: "Zur Theorie des Walzprocesses" (Stahl und Eisen, 1882, No. 7).

T. J. Stieltjes was one of the first mathematicians to address, from a very general point of view, the problem of the convergence of the so-called mechanical quadrature formulas. Recently, a very simple and elegant proof of Stieltjes' result for the case of continuous functions was given by J. V. Uspensky, who employs Weierstrass' fundamental theorem on the approximation of continuous functions by polynomials. However, in view of the importance of the question and some of the consequences arising from it, it seems to us not without interest to treat the problem without the aid of the aforementioned Weierstrass theorem and without preliminary study of a rather complex nature, and this will be the subject of the first section of this work.

The following proposals are the results of my research pertaining to the theory of Abelian integrals and functions with reducible periods.

The principle of correspondences between two points of Riemann surfaces can be regarded as a new form of the reduction problem, making it possible to apply transcendental methods of investigation. I shall indicate only the main results that I have managed to obtain in this direction.

In the summer of 1910, while participating as a lecturer in the geological mapping of Mount Vysokaya (the work was carried out by Prof. V. V. Nikitin at the invitation of the Board of the Nizhny Tagil Plants), I collected a collection of rocks. This may be of some interest for elucidating the composition of the constituent rocks, as well as the genesis of this large deposit of magnetic iron ore.

Since on a plane the analogous problem is solved very simply on the basis of Pascal's theorem, it is quite natural that the thoughts of geometers have been persistently directed toward finding a simple solution to the problem for space, and the lack of success has elevated this question to the status of a difficult problem. Pascal's theorem can be formulated in various ways, but, especially from the point of view of modern geometry, for which it served as one of the first foundations, this formulation must be connected with collineation, specifically one that transforms the conoprima determined by five points into itself. The construction based on a theorem analogous to that of Pascal is carried over into all geometric systems, particularly into the system of planes. However, the latter theorem, being of a non-positional character, is carried over only to related systems; for example, it is not carried over to the system of planes, just as the theorem of a non-positional character just cited is not applicable to the system of lines in a plane.

Let us turn to analogous constructions in space, which are a consequence of a theorem analogous to Pascal's theorem. Since the construction based on this theorem involves the construction of two hyperboloids of a linear prima, to which the required conosecund also belongs, and for this it is necessary to construct two hexaprimas, it is clear that the given data may consist of such tangents, together with their points of contact, as are sufficient for the construction of the hexaprimas.To understand why Pascal's theorem, and consequently its analogue, is of fundamental importance, it suffices to point out that these theorems are merely particular expressions of the deepest and most important fundamental theorem of new geometry, according to which, in two projective systems, linear aggregates correspond to linearaggregates, quadratic to quadratic, and, in general, aggregates of the n-th order correspond to aggregates of the same order. Moreover, intersections correspond to intersections, tangencies to tangencies, and involutions to involutions.

(Regarding Bowen's book "The later stages of the evolution of the igneous rocks"). The author of this note has long distinguished, among igneous rocks, between normal ones with more or less strict signs of chemical equilibrium and anomalous ones, which are not subject to any laws of equilibrium or any strict scientific classification, but bear clear signs of a sequential course of phenomena, the sequence of which, thanks to them, can be elucidated. Such rocks can only be described, and from the description, historical and geological dates can be deduced. Now, after the publication of Bowen's important work, one involuntarily inclines to the thought of the very weak representation of normal rocks, and perhaps even their absence; in nature, only approximations to them are represented, which is why they should rather be regarded not as normal, but as ideal.

In the article "Results of the First Stage of Experimental Research on the Structure of Crystals" (in the footnote on p. 361), a principle for determining the structure based on the parallelhedron of the smallest volume is established, which is equivalent to the principle of establishing a system of parallelhedra of the highest order permissible for a given regular system of points. A criterion for the correctness of constructing the parallelhedron can be the test of the possibility of a different, more specific, arrangement of one of the atoms represented in the chemical formula in the smallest number, or, even better, if there are no single atoms, the placement of that special point which occupies a unique position in the parallelhedron.

The first conclusion to be drawn from the observations made is the undeniable connection between the formation of actinolite and the decomposition of plagioclase. This connection is so inseparable that, as mentioned, augite decidedly never comes into contact with plagioclase pseudomorphs anywhere. We are, of course, unaware of the immediate primary cause of the decomposition of plagioclase and augite, and it undoubtedly stems from the still-unknown composition of the mineralized solution that penetrated the rock, which, however, must have contained both CaO (from the decomposition of plagioclase) and K₂O (otherwise muscovite would not have formed), with the first component diffusing from the plagioclase and the second diffusing toward it. And since at the points where the diffusing components exited the boundaries of the plagioclase, we see the transformation of augite into actinolite, and since CaO is the only component originating from the plagioclase whose content in actinolite is greater than in augite, we must conclude that it was precisely this component that was halted and absorbed by the augite, which was also affected by the overall chemical alteration of the rock—more pronounced in the plagioclase than in the augite.

Korkin and Zolotarev establish that the sought polynomials have n roots in the interval (-1, +1) and derive a series of equations that these roots satisfy; from the analysis of these equations, they deduce the uniqueness of the solution to the problem. This note proposes a simpler proof of the uniqueness of the solution, requiring only knowledge of the number of roots of the sought polynomials in the interval (-1, +1). The method of proof can be successfully applied to many similar problems of a more general nature.