Vol 5 Iss. 4-5

Precise speed control of engines and machines constitutes one of the most important tasks of modern mechanical engineering. This book aims to fill these gaps and represents an attempt at a systematic exposition of the practically applied theory of drum governors, primarily based on the author's own research and six years of experience in the design and construction of drum governors. The work covers in detail the following topics: centrifugal and drum governors, kinematics, statics, dynamics of drum governors, their design, and other issues.

If through any point on a quadratic cylinder we draw a secant plane (defining a conoprima k), a tangent plane, and in it some skew line d) (not in the plane of the curve k) and not a generator of the cylinder), and then from each point of this line draw two rays in the diametral plane of the cylinder through the points of the curve k), we obtain a ruled surface of the third order.

The described deposit is located in the Yekaterinburg District of Perm Province, in the northeast corner of block 109 of the Ayat Dacha of the Verkh-Isetsk District. The vein rock, of igneous origin, cutting through the thickness of the host rocks like dikes, is a rock of porphyritic structure, sometimes with a clear composition. The phenocrysts are large grains of albite, varying in size from No. 0 to No. 4. The groundmass consists of small crystals, usually elongated in the direction of flow, of the same albite, quartz, and small grains of orthoclase that cannot be optically investigated and are only detected by chemical analysis. Inclusions of apatite crystals, pyrite, and occasionally tourmaline and stibnite are observed almost everywhere. Based on the work carried out to date, the deposit as it currently appears has no commercial significance, but it must be kept in mind that many very essential aspects of it have not yet been clarified.

Systems of points and rays in the plane are, as is known, not related. However, related to a system of points are the secunds of conoprimas of points and rays having three fixed elements, and among those having two fixed elements, only the conoprimas of points (tetraprimas) are related. Now we will show that the secunds of parabolas of rays with two fixed rays are related to the system of rays in the plane.

The question of the existence of so-called 'fundamental functions' for higher-order differential equations has been the subject of research in a number of works, but undoubtedly further elaboration is possible in the sense of applying various methods to its solution. This article represents an attempt to generalize the method of the American geometer Max Mason, as expounded by him for second-order differential equations, to the case of fourth-order differential equations, to which, as is known, the question of the oscillations of an elastic inhomogeneous rod reduces.

The projective relations elucidated in the preceding note would achieve a high degree of clarity if it were possible to establish, by a simple construction, such a correlation between the conoprimas of a secund and the points of the plane that the extra-elements of the one correspond to the extra-points of the other. In the secund of pencils of rays, there is contained a prima of those which are represented by a pair of points (more precisely, by a pair of linear primas of rays). Since in each linear prima there are three such special conoprimas represented (of which one pair may be imaginary), it is perfectly obvious that the prima of special primas of rays is correlative to a curve of the third order.

According to Steiner's famous theory, two given involutions of pairs of points on lines in a plane determine an involution on any line in the plane, that is, the complete secund of involution. The determining factor of all these involutions is a linear prima of curves, namely a pencil of conics (Kegelschnittbüschel, according to Steiner), having two pairs of points in common, of which not only one, but both pairs can be imaginary. Each line intersects each curve of the pencil in a pair of points belonging to its involution. Specifically, we can define a collineation by two axes without any involutions. If we call the axes, whose points are the real double points of all involutions, real axes, and the axes of isotropic involutions imaginary axes, then we obtain that every axial collineation can be defined by a pair of axes, real or imaginary (see the article).

In the preceding note, we solved the problem of transferring all kinds of constructions carried out in the plane to the system of conoprimas of points and rays; and conversely, we reduced the solutions of all kinds of problems in the secunds of conoprimas of points and rays having three fixed elements to ordinary problems in the plane. The thought naturally arises to extend the solution also to those simplest cases where in the secunds of conoprimas there are only two or only one common element.

In any given conoprima, we can arbitrarily take two groups of points, four in each, and establish a general collinearity based on them (see the article). A cycle can consist of a different number of points, up to infinity. If, for example, a point is self-homologous in the collineation, then the entire cycle consists of a single point; if we have a double homology of points A and A', then the entire cycle reduces to two points, and so on. In the general case, the cycle encompasses a significant number of points, or even an infinite number, and it may happen that all points of the conoprima form part of a single cycle. If the collineation, established in some way from the given points, makes a conosecund self-homologous, then the problem of constructing the points of the latter reduces to a simple problem of collinear constructions.