scholarly journals A type of periodicity of certain quartic surfaces

1942 ◽  
Vol 7 (1) ◽  
pp. 73-80 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Present Note ◽  
Quartic Surface ◽  
Periodic Property ◽  
A Chain ◽  
Quartic Surfaces ◽  
Do So

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.

scholarly journals On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation

1926 ◽  
Vol 110 (756) ◽  
pp. 700-708
Keyword(s):  
Present Note ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Four Dimensions ◽  
Cubic Surfaces ◽  
Mutual Relations ◽  
Quartic Surfaces ◽  
Three Dimensional Space

There exists in space of four dimensions an interesting figure of 15 lines and 15 points, first considered by Stéphanos (‘Compt. Rendus,’ vol. 93, 1881), though suggested very clearly by Cremona’s discussion of cubic surfaces in three-dimensional space. In connection with the figure of 15 lines there arises a quartic surface, the intersection of two quadrics, which is of importance as giving rise by projection to the Cyclides, as Segre has shown in detail (‘Math. Ann.,’ vol. 24, 1884). The symmetry of the figure suggests, howrever, the consideration of 15 such quartic surfaces; and it is natural to inquire as to the mutual relations of these surfaces, in particular as to their intersections. In general, two surfaces in space of four dimensions meet in a finite number of points. It appears that in this case any two of these 15 surfaces have a curve in common; it is the purpose of the present note to determine the complete intersection of any two of these 15 surfaces. Similar results may be obtained for a system of cubic surfaces in three dimensions, corresponding to those here given for this system of quartic surfaces in four dimensions, since the surfaces have one point in common, which may be taken as the centre of a projection.

scholarly journals Matrices of integers associated with self-transformations of surfaces

1948 ◽  
Vol 193 (1032) ◽  
pp. 25-43
Keyword(s):  
Rational Curve ◽  
Pell Equation ◽  
Quartic Surface ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
The Matrix ◽  
Quartic Surfaces ◽  
Infinite Sequences ◽  
Square Matrix

Let c = ( c 1 , . . . , c r ) be a set of curves forming a minimum base on a surface, which, under a self-transformation, T , of the surface, transforms into a set T c expressible by the equivalences T c = Tc, where T is a square matrix of integers. Further, let the numbers of common points of pairs of the curves, c i , c j , be written as a symmetrical square matrix Г. Then the matrix T satisfies the equation TГT' = Г. The significance of solutions of this equation for a given matrix Г is discussed, and the following special surfaces are investigated: §§4-7. Surfaces, in particular quartic surfaces, wìth only two base curves. Self-transformations of these depend on the solutions of the Pell equation u 2 - kv 2 = 1 (or 4). §8. The quartic surface specialized only by being made to contain a twisted cubic curve. This surface has an involutory transformation determined by chords of the cubic, and has only one other rational curve on it, namely, the transform of the cubic. The appropriate Pell equation is u 2 - 17 v 2 = 4. §9. The quartic surface specialized only by being made to contain a line and a rational curve of order m to which the line is ( m - 1)⋅secant (for m = 1 the surface is made to contain two skew lines). The surface has two infinite sequences of self-transformations, expressible in terms of two transformations R and S , namely, a sequence of involutory transformations R S n , and a sequence of non-involutory transformations S n .

scholarly journals The geometrical construction of Maschke's quartic surfaces

1945 ◽  
Vol 7 (2) ◽  
pp. 93-103 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Finite Order ◽  
Geometrical Construction ◽  
Remarkable Property ◽  
Quartic Surface ◽  
Quartic Form ◽  
Quartic Surfaces

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

scholarly journals Conics on a Maschke Surface

1946 ◽  
Vol 7 (3) ◽  
pp. 153-161
Author(s):  
W. L. Edge
Keyword(s):  
Quartic Surface ◽  
Linear Complexes ◽  
Image Position ◽  
Quartic Surfaces

The six quaternary quartic formswere first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

Configurations defined by six lines in space of three dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 52-68 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Linear System ◽  
Classical Problem ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Lines In Space ◽  
Pass Through ◽  
Quartic Surfaces

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.

scholarly journals On degrees and genera of curves on smooth quartic surfaces in P3

1984 ◽  
Vol 96 ◽  
pp. 127-132 ◽  
Author(s):  
Shigefumi Mori
Keyword(s):  
Singular Curve ◽  
Quartic Surface ◽  
Quartic Surfaces

Our result is motivated by the results [GP] of Gruson and Peskin on characterization of the pair of degree d and genus g of a non-singular curve in P3. In the last step, they construct the required curve C on a singular quartic surface when Here we consider curves on smooth quartic surfaces.

Complexes of Conics and the Weddle Surface

1924 ◽  
Vol 22 (3) ◽  
pp. 201-216
Author(s):  
C. G. F. James
Keyword(s):  
Arbitrary Point ◽  
Three Dimensions ◽  
Irreducible Curve ◽  
Quartic Surface ◽  
Ordinary Space ◽  
Four Dimensions ◽  
Interesting Connection ◽  
Linear Complexes ◽  
Pass Through ◽  
Do So

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.

Theoretical Efficiency of a Cranked Link Chain Drive

1986 ◽  
Vol 200 (5) ◽  
pp. 375-377 ◽  
Author(s):  
N E Hollingworth ◽  
D A Hills
Keyword(s):  
Present Note ◽  
Contact Forces ◽  
Chain Drive ◽  
Theoretical Evaluation ◽  
Chain Transmission ◽  
A Chain

In a previous paper (1), the contact forces in a chain bearing during articulation were established. The present note describes the application of these results to the theoretical evaluation of efficiency for a conventional chain transmission using cranked link (or offset) type chain.

scholarly journals II. Notes on physical geology. No. VII. On the secular inequalities in terrestrial climates depending on the perihelion longitude and eccentricity of the earth’s orbit

1881 ◽  
Vol 31 (206-211) ◽  
pp. 473-477
Keyword(s):  
Present Note ◽  
Mathematical Form ◽  
Physical Geology ◽  
Do So

The attention of geologists was first called by M. Adhémar, and afterwards more fully by Mr. James Croll, to the possible importance of these long inequalities in climate, in explaining the climates of geological periods, which differ considerably from those of the present time in the same places; but, so far as I know, no one has written, down these inequalities in a mathematical form, or calculated numerically the effects upon climate they are capable of producing. I shall attempt to do so in the present note.

scholarly journals Diagonal quartic surfaces and transcendental elements of the Brauer group

2010 ◽  
Vol 9 (4) ◽  
pp. 769-798 ◽  
Author(s):  
Evis Ieronymou
Keyword(s):  
Function Field ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Weak Approximation ◽  
Brauer Group ◽  
Quartic Surface ◽  
Simple Algebras ◽  
Central Simple ◽  
Quartic Surfaces ◽  
Torsion Part

ABSTRACTWe exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

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