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.

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 Changes in Incisor Third-Order Inclination Resulting from Vertical Variation in Lingual Bracket Placement

2009 ◽  
Vol 79 (4) ◽  
pp. 747-754 ◽  
Author(s):  
Michael Knösel ◽  
Klaus Jung ◽  
Liliam Gripp-Rudolph ◽  
Thomas Attin ◽  
Rengin Attin ◽  
...  
Keyword(s):  
Null Hypothesis ◽  
Interindividual Variation ◽  
Occlusal Plane ◽  
Third Order ◽  
Vertical Variation ◽  
Independent Variables ◽  
The Third ◽  
Dependent Variables ◽  
Incisor Inclination ◽  
The Impact

Abstract Objective: To test the null hypothesis that third-order measurements are not correlated to lingual incisor features seen on radiographs. Material and Methods: The lateral headfilms of 38 untreated, norm-occlusion subjects without incisor abrasions or restorations were used for third-order measurements of upper and lower central incisors and assessment of the inclination of four sites suitable for lingual bracket placement with reference to the occlusal plane perpendicular. Lingual sections were determined by the tangents at the incisal fossa (S1), at the transition plateau between incisal fossa and the cingulum (S2), by a constructed line reaching from the incisal tip to the cingulum (S3), and by a tangent at the cingulum convexity (S4). Third-order angles were also assessed on corresponding dental casts using an incisor inclination gauge. Regression analysis was performed using the third-order measurements of both methods as the dependent variables and the inclination of the lingual enamel sections (S1, S2, S3, S4) as the independent variables. Results: The null hypothesis was rejected. For the most common bracket application sites located on the lingual shovel (S1 and S2), third-order inclination changes of 0.4–0.7 degrees are expected for each degree of change in the inclination of the lingual surface. The impact of bracket placement errors on third-order angulation is similar between sections S1 and S2 and the cingulum convexity (S4). Section S3 proved to be least affected by interindividual variation. Conclusion: The third-order measurements are correlated to lingual incisor features. Accordingly, third-order changes resulting from variation in lingual bracket placement can be individually predicted from radiographic assessments.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

Strongly Oscillatory and Nonoscillatory Subspaces of Linear Equations

1975 ◽  
Vol 27 (1) ◽  
pp. 106-110 ◽  
Author(s):  
J. Michael Dolan ◽  
Gene A. Klaasen
Keyword(s):  
Linear Equation ◽  
Linear Space ◽  
Nontrivial Solution ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Oscillatory Solutions ◽  
The Third ◽  
All Solutions

Consider the nth order linear equationand particularly the third order equationA nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

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 On A Darboux Problem for A Third Order Hyperbolic Equation with Multiple Characteristics

1995 ◽  
Vol 2 (5) ◽  
pp. 469-490
Author(s):  
O. Jokhadze
Keyword(s):  
Hyperbolic Equation ◽  
Type Problem ◽  
Darboux Problem ◽  
Third Order ◽  
Independent Variables ◽  
The Third ◽  
Multiple Characteristics ◽  
Order Hyperbolic Equation ◽  
Darboux Type Problem ◽  
Stability Of The Solution

Abstract A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space , α ≥ 0, is introduced where the problem under consideration is investigated. The real number α 0 is found such that for α > α 0 the problem is solved uniquely and for α < α 0 it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.

Direct contribution of the surface layers to the Earth's dynamical flattening

2007 ◽  
Vol 3 (S248) ◽  
pp. 403-404
Author(s):  
Y. Liu ◽  
C. L. Huang
Keyword(s):  
Mixed Data ◽  
Real Surface ◽  
Surface Layers ◽  
Third Order ◽  
Order Accuracy ◽  
Positive Effects ◽  
The Third ◽  
Global Dynamic ◽  
The Earth ◽  
Topography Data

AbstractThe global dynamic flattening (H) is an important quantity in research of rotating Earth. Precession observations give Hobs = 0.0032737 ≈ 1/305.5. We recalculate the geometrical flattening profile of the Earth interior from potential theory in hydrostatic equilibrium. Results coincide with that of Denis (1989). We derive expression for H to the third-order accuracy and obtain HPREM = 1/308.5. This matches similar studies, in which there is a difference about 1% between this and the observed value. In order to understand where this difference comes from, we replace the homogenous outermost crust and oceanic layers in PREM with some real surface layers data, such as oceanic layer (ECCO), topography data (GTOPO30), crust data (CRUST2.0) and mixed data (ETOPO5). Our results deviate from the observed value more than HPREM. These results verify the isostasy theory indirectly and may imply that the “positive” effects from such as mantle circulation associated with the density anomalies maybe larger than thought before.

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 Comparison theorems of Hille-Wintner type for third order linear differential equations

1980 ◽  
Vol 21 (2) ◽  
pp. 175-188 ◽  
Author(s):  
L. Erbe
Keyword(s):  
Differential Equations ◽  
Linear Equation ◽  
Linear Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Third Order ◽  
Order Linear ◽  
The Third ◽  
Linear Differential

Integral comparison theorems of Hille-Wintner type of second order linear equations are shown to be valid for the third order linear equation y‴ + q(t)y = 0.

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