STANDARD NOMOGRAPHIC FORMS FOR EQUATIONS IN THREE VARIABLES

1935 ◽  
Vol 12 (1) ◽  
pp. 14-40 ◽  
Author(s):  
F. M. Wood
Keyword(s):  
Unit Circle ◽  
Fourth Order ◽  
Third Order ◽  
Rectangular Hyperbola ◽  
Practical Utility ◽  
The Third ◽  
Cubic Curve ◽  
Final Adjustment ◽  
Straight Lines ◽  
Suitable Location

Equations of the third and fourth nomographic order in three variables have been dealt with and classified. Equations of the third order may be reduced to one of two standard forms, α + β + γ = 0 and α + βγ = 0, which give alignment charts composed of three straight lines. Equations of the fourth order may also be reduced to one of two standard forms, resulting in charts composed of (a) two straight lines and a curve, or (b) two scales on a conic, and the third on another curve. Transformations of these four standard forms are given which permit of rapid and easy adjustment of the position and length of the scales for any given example, resulting in a chart of practical utility. Although the underlying theory has been studied by other writers, notably Soreau and Clark, it has possibly never appeared before in such a neat form. On this account, and also because of the standard transformations, it is felt that this article is of particular value.Standard forms have also been developed for third order equations leading to charts composed of two scales on a conic and a third straight scale, and in conclusion a third type of chart, in which all three scales appear on a single cubic curve, has been standardized. The practical value of the last type is questionable, but the conic charts are of use since we may arbitrarily choose the unit circle, or the rectangular hyperbola, for our conic scales. Final adjustment forms which permit suitable location of the scales in particular examples have been obtained in every case.

Properties of Dupin Cyclide and Their Application. Part 3: The Problem of Apollonius

2015 ◽  
Vol 3 (4) ◽  
pp. 3-14 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Practical Application ◽  
Practical Utility ◽  
Engineering Graphics ◽  
The Third ◽  
Focal Surface ◽  
Dupin Cyclide ◽  
Straight Lines

In the first part of work was addressed mainly the issue of properties under Dupin cyclide, and given some examples of their applications: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of Dupin cyclid. The second part of work continued with consideration of the use of property under a lie of Dupin. It is determined that the focal surface of cyclid of Dupin is degenerated in the lines and represent curves of the second order. Here under a lie can be defined conic curve and a sphere whose center lies on the focal curve. Polyconic conformity these focal curves is revealed. The article show the formation of the surface of the fourth order on the basis of defocusing curves of the second order. In this issue of the journal the reader is invited to consider the practical application of properties under a lie of Dupin for example of well known problems with on-voltage. If the first part of the work was cited only three ways of solving the problem of Apollonius, in the third part the author considers other possible mates: as at zero the size of the radius of the circle and the demon is of course great. All decisions – both known and not really based on properties of Dupin cyclide. In the course of engineering graphics, introductory tests, as they say now, drawing on architectural faculties there are tasks, dedicated to the mating arcs of circles with straight lines, and circles passing through the points in various combinations. Therefore, the proposed practical application cannot be considered far-fetched – it is based on the practical utility of method.

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.

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

scholarly journals Volume derivatives of the Grüneisen parameter in the limit of infinite pressure

2019 ◽  
Vol 97 (1) ◽  
pp. 114-116 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Fundamental Theorem ◽  
High Pressures ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third ◽  
Properties Of Materials ◽  
Derivatives Of

Expressions have been obtained for the volume derivatives of the Grüneisen parameter, which is directly related to the thermal and elastic properties of materials at high temperatures and high pressures. The higher order Grüneisen parameters are expressed in terms of the volume derivatives, and evaluated in the limit of infinite pressure. The results, that at extreme compression the third-order Grüneisen parameter remains finite and the fourth-order Grüneisen parameter tends to zero, have been used to derive a fundamental theorem according to which the volume derivatives of the Grüneisen parameter of different orders, all become zero in the limit of infinite pressure. However, the ratios of these derivatives remain finite at extreme compression. The formula due to Al’tshuler and used by Dorogokupets and Oganov for interpolating the Grüneisen parameter at intermediate compressions has been found to satisfy the boundary condition at infinite pressure obtained in the present study.

New bounds on effective elastic moduli of two-component materials

1982 ◽  
Vol 380 (1779) ◽  
pp. 305-331 ◽  
Keyword(s):  
Shear Modulus ◽  
Bulk Modulus ◽  
Fourth Order ◽  
Geometrical Parameters ◽  
Effective Moduli ◽  
Third Order ◽  
Effective Shear Modulus ◽  
The Third ◽  
Two Component ◽  
Order Bounds

Based on the perturbation solution, we derive new bounds on the effective moduli of a two-component composite material which are exact up to fourth order in δ μ = μ 1 — μ 2 and δ K = K 1 — K 2 , where μ i and K i , i = 1, 2, are the shear and bulk modulus, respectively, of the phases. The bounds on the effective bulk modulus involve three microstructural parameters whereas eight parameters are needed in the bounds on the effective shear modulus. For engineering calculations, we recommend the third-order bounds on the effective shear modulus which require only two geometrical parameters. We show in detail how Hashin-Shtrikman’s bounds can be extended and how Walpole’s bounds can be improved using two inequalities on the two geometrical parameters that appear in the third-order bounds on the effective shear modulus. The third- and fourth-order bounds on the effective moduli are shown to be more restrictive than, or at worst, coincident with, existing bounds due to Hashin and Shtrikman, McCoy, Beran and Molyneux and Walpole.

scholarly journals SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY

2020 ◽  
Vol 60 (2) ◽  
pp. 98-110
Author(s):  
Amlan Kanti Halder ◽  
Andronikos Paliathanasis ◽  
Peter Gavin Lawrence Leach
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Singularity Analysis ◽  
Travelling Wave ◽  
Similarity Solutions ◽  
Third Order ◽  
The Third ◽  
Beam Equations ◽  
Complete Study ◽  
Wave Reduction

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

The mechanism of the Cannizzaro reaction of Formaldehyde

1954 ◽  
Vol 7 (4) ◽  
pp. 335 ◽  
Author(s):  
RJL Martin
Keyword(s):  
Fourth Order ◽  
Kinetic Data ◽  
Order Reaction ◽  
Rate Equations ◽  
Third Order ◽  
The Third ◽  
Hydride Ion ◽  
Cannizzaro Reaction ◽  
Wide Range ◽  
Mechanism Of The Reaction

For a wide range of concentrations of formaldehyde and alkali, the Cannizzaro reaction of formaldehyde can be described as the sum of a third and a fourth order reaction. However, the concentrations which are used for the rate equations must be corrected for the amount of methylene glycol anion present. The dissociation constant of methylene glycol as determined from the kinetic data is the same magnitude as that derived electrometrically. The mechanism of the reaction is interpreted as a reaction between formaldehyde and the hydride ion donors CH2(O-)(OH) and CH2(O-)(O-) It is shown why the third order reaction proposed by previous workers is not always applicable.

New bounds on the effective thermal conductivity of N -phase materials

1982 ◽  
Vol 380 (1779) ◽  
pp. 333-348 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Fourth Order ◽  
Third Order ◽  
Two Phase ◽  
The Third ◽  
Component Material ◽  
Order Bounds ◽  
Correlation Parameters ◽  
Two Parameters

Based on the perturbation solution to the effective thermal conductivity problem for an N -component material, we derive new third- and fourth-order bounds for the effective thermal conductivity of the composite. The new third-order bounds are accurate to third order in | ϵ a — ϵ b |, where ϵ a is the thermal conductivity of phase a and require a total of ½ N ( N — 1) 2 third-order correlation parameters in their evaluation. For two-phase composites, these bounds require only one geometrical parameter and are identical to Beran’s bounds. For N ≽ 3, the third-order bounds use the same information as Beran’s bounds, but always more restrictive than Beran’s bounds. The new fourth-order bounds are accurate to fourth-order in | ϵ a - ϵ b | and require an additional ¼ N 2 (N — 1) 2 fourth-order correlation parameters. For N = 2, only two parameters are needed, and the fourth-order bounds are shown to be more restrictive than the third-order Beran’s bounds and the second-order Hashin-Shtrikman’s bounds. The applica­tion of the third-order bounds to a spherical cell material of Miller is illustrated.

scholarly journals Bounds on the conductivity of a random array of cylinders

1988 ◽  
Vol 417 (1852) ◽  
pp. 59-80 ◽  
Keyword(s):  
Fourth Order ◽  
Volume Fraction ◽  
Effective Conductivity ◽  
Basis Set ◽  
Circular Cylinders ◽  
Third Order ◽  
Entire Volume ◽  
The Third ◽  
Wide Range ◽  
Order Bounds

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity σ e of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity σ 2 randomly distributed throughout a matrix of conductivity σ 1 . Both bounds involve the microstructural parameter ξ 2 which is an integral that depends upon S 3 , the three-point probability function of the composite (G. W. Milton, J. Mech. Phys. Solids 30, 177-191 (1982)). The key multidimensional integral ξ 2 is greatly simplified by expanding the orientation-dependent terms of its integrand in Chebyshev polynomials and using the orthogonality properties of this basis set. The resulting simplified expression is computed for an equilibrium distribution of rigid cylinders at selected ϕ 2 (cylinder volume fraction) values in the range 0 ≼ ϕ 2 ≼ 0.65. The physical significance of the parameter ξ 2 for general microstructures is briefly discussed. For a wide range of ϕ 2 and α = σ 2 /σ 1 , the third-order bounds significantly improve upon second-order bounds which only incorporate volume fraction information; the fourth-order bounds, in turn, are always more restrictive than the third-order bounds. The fourth-order bounds on σ e are found to be sharp enough to yield good estimates of σ e for a wide range of ϕ 2 , even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting ( α = ∞), moreover, the fourth-order lower bound on σ e provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i. e. up to a volume fraction of 65%.

scholarly journals XX. A memoir on curves of the third order

1857 ◽  
Vol 147 ◽  
pp. 415-446 ◽  
Keyword(s):  
Class I ◽  
Geometrical Theory ◽  
Third Order ◽  
The Third ◽  
Cubic Curve ◽  
The Subject ◽  
Definition Of

A curve of the third order, or cubic curve, is the locus represented by an equation such as U=(*)( x , y , z ) 3 =0; and it appears by my “Third Memoir on Quantics,” that it is proper to consider, in connexion with the curve of the third order U = 0, and its Hessian HU=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of U, and therefore, when the variables x , y , z of U are considered as point coordinates, the variables ξ, η, ζ of PU and QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. A geometrical definition of the Pippian was readily found; the curve is in fact Steiner’s curve R 0 mentioned in the memoir “Allgemeine Eigenschaften der algebraischen Curven,” Crelle , t. xlvii. pp. 1-6, in the particular case of a basis-curve of the third order; and I also found that the Pippian might be considered as occurring implicitly in my “Mémoire sur les Courbes du Troisiéme Ordre,” Liouville , t. ix. p. 285, and “Nouvelles Remarques sur les Courbes du Troisiéme Ordre,” Liouville , t. x. p. 102. As regards the Quippian, I have not succeeded in obtaining a satisfactory geometrical definition; but the search after it led to a variety of theorems, relating chiefly to the first-mentioned curve, and the results of the investigation are contained in the present memoir. Some of these results are due to Mr. Salmon, with whom I was in correspondence on the subject. The character of the results makes it diflicult to develope them in a systematic order; hut the results are given in such connexion one with another as I have been able to present them in. Considering the object of the memoir to be the establishment of a distinct geometrical theory of the Pippian, the leading results will be found summed up in the nine different definitions or modes of generation of the Pippian, given in the concluding number. In the course of the memoir I give some further developments relating to the theory in the memoirs in Liouville above referred to, showing its relation to the Pippian, and the analogy with theorems of Hesse in relation to the Hessian. Article No. 1.— Definitions , &c . 1. It may be convenient to premise as follows:—Considering, in connexion with a curve of the third order or cubic, a point , we have— ( a ) The first or conic polar of the point. ( b ) The second or line polar of the point. The meaning of these terms is well known, and they require no explanation.

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