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.

scholarly journals II. On a special form of the general equation of a cubic surface and on a diagram representing the twenty-seven lines on the surface

1894 ◽  
Vol 185 ◽  
pp. 37-69
Keyword(s):  
Special Form ◽  
General Equation ◽  
Mathematical Journal ◽  
Third Order ◽  
Cubic Surface ◽  
The Third ◽  
Straight Lines

The existence of straight lines on a cubic surface, the number of them, and their relations to each other was first discussed in a correspondence between Salmon and Cayley. In a paper which appeared in 1849, in vol. 4 of the ‘Cambridge and Dublin Mathematical Journal,’ “On the Triple Tangent Planes of Surfaces of the Third Order,” Cayley gave a sketch of what was then known, and gave the equations of the forty-five planes in which the twenty-seven lines on the surface lie by threes, when the equation of the surface is taken in a particular form.

scholarly journals X. On the distribution of surfaces of the third order into species, reference to the absence or presence of singular points, and the reality of their lines

1863 ◽  
Vol 153 ◽  
pp. 193-241 ◽  
Keyword(s):  
Linear Equations ◽  
Ruled Surface ◽  
Real Surface ◽  
Real Point ◽  
Third Order ◽  
Cubic Surface ◽  
Independent Variables ◽  
The Third ◽  
Fourth Class ◽  
Rectangular Coordinates

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. vol. 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. Schläfli’s paper, "An attempt to determine the 27 lines &c.,” Quart. Math. Journ. vol. ii. pp. 56-65, and 110-120. This theory is reproduced and developed in the present memoir under the heading, I. General cubic surface of the third order and twelfth class; but the greater part of the memoir relates to the singular forms which are here first completely enumerated, and are considered under the headings II., III. &c. to XXII., viz. II. Cubic surface with a proper node, and therefore of the tenth class, &c., down to XXII. Ruled surface of the third order. Each of these families is discussed generally (that is, without regard to reality or non-reality), by means of a properly selected canonical form of equation; and for the most part, or in many instances, the reciprocal equation (or equation of the surface in plane-coordinates) is given, as also the equation of the Hessian surface and those of the Spinode curve; and it is further discussed and divided into species according to the reality or non-reality of its lines and planes. The following synopsis may be convenient :— I. General cubic surface, or surface of the third order and twelfth class. Species I. 1, 2, 3, 4, 5. II. Cubic surface with a proper node, and therefore of the tenth class. Species II. 1, 2, 3, 4, 5. III. Cubic surface of the ninth class with a biplanar node. Species III. 1, 2, 3, 4. IV. Cubic surface of the eighth class with two proper nodes. Species IV. 1, 2, 3, 4, 5, 6. V. Cubic surface of the eighth class with a biplanar node. Species V. 1, 2, 3, 4. VI. Cubic surface of the seventh class with a biplanar and a proper node. Species VI. 1, 2. VII. Cubic surface of the seventh class with a biplanar node. Species VII. 1, 2. VIII. Cubic surface of the sixth class with three proper nodes. Species VIII. 1, 2, 3, 4. IX. Cubic surface of the sixth class with two biplanar nodes. Species IX. 1,2, 3,4. X. Cubic surface of the sixth class with a biplanar and a proper node. Species X. 1, 2. XI. Cubic surface of the sixth class with a biplanar node. Species XI. 1, 2. XII. Cubic surface of the sixth class with a uniplanar node. Species XII. 1, 2. XIII. Cubic surface of the fifth class with a biplanar and two proper nodes. Species X III. 1, 2. XIV. Cubic surface of the fifth class with a biplanar node and a proper node. Species XIV. 1. XV. Cubic surface of the fifth class with a uniplanar node. Species XV. 1. XVI. Cubic surface of the fourth class with four proper nodes. Species XVI. 1, 2, 3. XVII. Cubic surface of the fourth class with two biplanar and one proper node. Species XVII. 1, 2, 3. XVIII. Cubic surface of the fourth class with one biplanar and two proper nodes. Species XVIII. 1. XIX. Cubic surface of the fourth class with a biplanar and a proper node. Species XIX. 1. XX. Cubic surface of the fourth class with a uniplanar node. Species XX. 1. XXI. Cubic surface of the third class with three biplanar nodes. Species XXI. 1, 2. XXII. Ruled surface of the third order and the third class. Species XXII. 1, 2, 3.—A. C. I. General cubic surface, or surface of the third order and twelfth class. Art. 1. As the system of coordinates undergoes various transformations (sometimes imaginary ones), it becomes necessary to adhere to an invariable system of a real meaning, for instance the usual one of three rectangular coordinates. We shall call this the system of fundamental cordinates , and define it by the condition that the coordinates of every real point (or the ratios of them, if they be four in number) shall be real. Consequently any system of rational and integral equations, expressed in variables of a real meaning, and where all the coefiicients are real, will be termed a real system (of equations), whether there be real solutions or none, provided that the number of equations do not exceed that of the variables, or of the quantities to be determined. The degree of the system will be the number of solutions of it when augmented by a sufficient number of arbitrary linear equations; and such degree will generally be the product of the degrees of the single equations. It is obvious that the system, whenever its degree is odd , represents a real continuum of as many dimensions are independent variables; for instance, every real quaternary cubic represents a real surface.

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.

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