On the Distribution of Surfaces of the Third Order into Species, in reference to the absence or presence of Singular Points, and the reality of their Lines

1953 ◽  
pp. 304-362
Author(s):  
L. Kollros
Keyword(s):  
Singular Points ◽  
Third Order ◽  
The Third

scholarly journals VII. A memoir on cubic surfaces

1869 ◽  
Vol 159 ◽  
pp. 231-326 ◽  
Keyword(s):  
Singular Points ◽  
Third Order ◽  
Cubic Surface ◽  
The Third ◽  
Cubic Surfaces ◽  
Present Subject

The present Memoir is based upon, and is in a measure supplementary to that by Pro­fessor Schläfli, “On the Distribution of Surfaces of the Third Order into Species, in reference to the presence or absence of Singular Points, and the reality of their Lines,” Phil. Trans, vol. cliii. (1863) pp. 193—241. But the object of the Memoir is different. I disregard altogether the ultimate division depending on the reality of the lines, attend­ing only to the division into (twenty-two, or as I prefer to reckon it) twenty-three cases depending on the nature of the singularities. And I attend to the question very much on account of the light to be obtained in reference to the theory of Reciprocal Surfaces. The memoir referred to furnishes in fact a store of materials for this purpose, inasmuch as it gives (partially or completely developed) the equations in plane-coordinates of the several cases of cubic surfaces, or, what is the same thing, the equations in point-coor­dinates of the several surfaces (orders 12 to 3) reciprocal to these repectively. I found by examination of the several cases, that an extension was required of Dr. Salmon’s theory of Reciprocal Surfaces in order to make it applicable to the present subject ; and the preceding “Memoir on the Theory of Reciprocal Surfaces” was written in connexion with these investigations on Cubic Surfaces. The latter part of the Memoir is divided into sections headed thus:— “Section I = 12, equation (X, Y, Z, W ) 3 = 0” &c. referring to the several cases of the cubic surface; but the paragraphs are numbered continuously through the Memoir. The twenty-three Cases of Cubic Surfaces—Explanations and Table of Singularities . Article Nos. 1 to 13. 1. I designate as follows the twenty-three cases of cubic surfaces, adding to each of them its equation:

scholarly journals III. A memoir on cubic surfaces

1869 ◽  
Vol 17 ◽  
pp. 221-222
Keyword(s):  
Singular Points ◽  
Third Order ◽  
Cubic Surface ◽  
The Third ◽  
Cubic Surfaces ◽  
Present Subject

The present Memoir is based upon, and is in a measure supplementary to that by Professor Schläfli, “On the Distribution of Surfaces of the Third Order into Species, in reference to the presence or absence of Singular Points, and the reality of their Lines,” Phil. Trans, vol. cliii. (1863) pp. 193–241. But the object of the Memoir is different. I disregard altogether the ultimate division depending on the reality of the lines, attending only to the division into (twenty-two, or as I prefer to reckon it) twenty-three cases depending on the nature of the singularities. And I attend to the question very much on account of the light to be obtained in reference to the theory of Reciprocal Surfaces. The memoir referred to furnish.es in fact a store of materials for this purpose, inasmuch as it gives (partially or completely developed^the equations in plane-coordinates of the several cases of cubic surfaces; or, what is the same thing, the equations in point-coordinates of the several surfaces (orders 12 to 3) reciprocal to these respectively. I found by examination of the several cases, that an extension was required of Dr. Salmon’s theory of Reciprocal Surfaces in order to make it applicable to the present subject; and the preceding “Memoir on the Theory of Reciprocal Surfaces” was written in connexion with these investigations on Cubic Surfaces. The latter part of the Memoir is divided into sections headed thus:—“Section 1 = 12, equation (X, Y, Z, W) 3 = 0” &c. referring to the several cases of the cubic surface; but the paragraphs are numbered continuously through the Memoir. The principal results are included in the following Table of singularities. The heading of each column shows the number and character of the case referred to, viz. C denotes a conic node, B a biplanar node, and U a uniplanar node; these being further distinguished by subscript numbers, showing the reduction thereby caused in the class of the surface: thus XIII=12—B 3 —2 C 2 indicates that the case XIII is a cubic surface, the class whereof is 12—7, = 5, the reduction arising from a biplanar node, B 4 , reducing the class by 3, and from 2 conic nodes, C 2 , each reducing the class by 2.

III. Distribution of the surface of the third order into species, in reference to the absence or presence of singular points, and the reality of its lines

1863 ◽  
Vol 12 ◽  
pp. 327-329
Keyword(s):  
Singular Point ◽  
Singular Points ◽  
Third Order ◽  
The Third ◽  
General Surface

The theory of the 27 lines on a surface of the third order is due to Mr. Cayley and Dr. Salmon; and the effect as regards the 27 lines of a singular point or points on the surface, was first considered by Dr. Salmon in the paper “On the triple tangent planes of a surface of the third order,” Camb. and Dub. Math. Journ. t. iv. pp. 252–260 (1849). The theory as regards the reality or non-reality of the lines on a general surface of the third order, is discussed in Dr. Schläffle’s Paper, “An attempt to determine the 27 lines, &c.,” Quart. Math. Journ. t. ii. pp. 56-65, and 110—120.

scholarly journals СИЛОВИЙ АНАЛІЗ МЕХАНІЗМУ КОЛИВАЛЬНОГО РУХУ ВУШКОВИХ ГОЛОК ОСНОВОВ’ЯЗАЛЬНОЇ МАШИНИ

2019 ◽  
Vol 134 (3) ◽  
pp. 26-35
Author(s):  
В. М. Дворжак
Keyword(s):  
Numerical Method ◽  
Analytical Method ◽  
Singular Points ◽  
Light Industry ◽  
Planar Mechanisms ◽  
Third Order ◽  
The Third ◽  
Vector Algebra ◽  
Force Calculation

Improving methods of designing technological machines mechanisms of light industry in CAD-programs. Methodology. Used vector algebra apparatus; analytical method for the force calculation of planar mechanisms  of  the  third  class  of  the  third  order  on  the  basis  of  the  method  of  vector  transformation  of coordinates; a numerical method for calculating the position functions of characteristic and singular points of a third-class third-order mechanism with rotating kinematic pairs.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Sign in / Sign up

Export Citation Format

Share Document