XXIV. On the tangential of a cubic

1858 ◽  
Vol 148 ◽  
pp. 461-463
Keyword(s):  
Canonical Form ◽  
Second Line ◽  
Third Order ◽  
Cubic Form ◽  
The Third ◽  
Cubic Curve ◽  
Right Hand ◽  
The Right ◽  
Cubic Forms

In my “Memoir on Curves of the Third Order, ”I had occasion to consider a derivative which may be termed the "tangential” of a cubic, viz. the tangent at the point ( x , y , z ) of the cubic curve (*)( x , y , z ) 3 = 0 meets the curve in a point ( ξ , n , ζ ) which is the tangential of the first-mentioned point; and I showed that when the cubic is represented in the canonical form x 3 + y 3 + z 3 +6 lxyz = 0 , the coordinates of the tangential may be taken to be x ( y 3 — z 3 ) : y ( z 3 — x 3 ) : z ( x 3 — y 3 ). The method given for obtaining the tangen­tial may be applied to the general form ( a , b , c , f , i , j , k , l )( x , y , z ) 3 : it seems desirable, in reference to the theory of cubic forms, to give the expression of the tangential for the general form; and this is what I propose to do, merely indicating the steps of the calculation, which was performed for me by Mr. Creedy. The cubic form is ( a , b , c , f , g , h , i , j , k , l )( x , y , z ) 3 , which means ax 3 + by 3 + cz 3 +3 fy 2 z +3 gz 2 x +3 hx 2 y +3 iyz 2 +3 jzx 2 +3 hxy 2 +6 lxyz ; and the expression for ξ is obtained from the equation x 2 ξ =( b , f , i , c )( j , f , c , i , g , l )( x , y , z ) 2 ,—( h , b , i , f , l , k )( x , y , z ) 2 ) 3 — ( a , b , c , f , g , h , i , j , k , l )( x , y , z ) 3 (C x +D), where the second line is in fact equal to zero, on account of the first factor, which vanishes. And C, D, denote respectively quadric and cubic functions of ( y , z ), which are to be determined so as to make the right-hand side divisible by x 2 the resulting value of ξ may be modified by the adjunction of the evanescent term

IV. On the tangential of a cubic

1859 ◽  
Vol 9 ◽  
pp. 167-167
Keyword(s):  
Canonical Form ◽  
Third Order ◽  
The Third ◽  
Cubic Curve ◽  
Cubic Forms

In my "Memoir on Curves of the Third Order,” Phil. Trans, vol. cxlvii. (1857), I had occasion to consider a derivative which may be termed the “tangential” of a cubic, viz. the tangent at the point ( x, y, z ) of the cubic curve (*≬ x, y, z ) 3 =0 meets the curve in a point ( ε, η, ζ ), which is the tangential of the first-mentioned point; and I showed that when the cubic is represented in the canonical form x 3 + y 3 + z 3 +6 Ixyz = 0, the coordinates of the tangential may be taken to be x ( y 3 – z 3 ) : y ( z 3 – x 3 ) : z ( x 3 –y 3 ). The method given for obtaining the tangential may be applied to the general form ( a, b, c,f, g, h, i, j, k, l ≬ x , y, z ) 3 : it seems desirable, in reference to the theory of cubic forms, to give the expression of the tangential for the general form; and this is what I propose to do, merely indicating the steps of the calculation, which was performed for me by Mr. Creedy.

Eosinophilic vasculitis: an inhabitual and resistant manifestation of a vasculitis

VASA ◽  
2010 ◽  
Vol 39 (4) ◽  
pp. 344-348 ◽  
Author(s):  
Jandus ◽  
Bianda ◽  
Alerci ◽  
Gallino ◽  
Marone
Keyword(s):  
Skin Biopsy ◽  
Intravenous Iron ◽  
Small Vessel Vasculitis ◽  
High Dose ◽  
Raynaud Phenomenon ◽  
The Third ◽  
Left Elbow ◽  
Right Hand ◽  
The Right ◽  
Trunk Skin

A 55-year-old woman was referred because of diffuse pruritic erythematous lesions and an ischemic process of the third finger of her right hand. She was known to have anaemia secondary to hypermenorrhea. She presented six months before admission with a cutaneous infiltration on the left cubital cavity after a paravenous leakage of intravenous iron substitution. She then reported a progressive pruritic erythematous swelling of her left arm and lower extremities and trunk. Skin biopsy of a lesion on the right leg revealed a fibrillar, small-vessel vasculitis containing many eosinophils.Two months later she reported Raynaud symptoms in both hands, with a persistent violaceous coloration of the skin and cold sensation of her third digit of the right hand. A round 1.5 cm well-delimited swelling on the medial site of the left elbow was noted. The third digit of her right hand was cold and of violet colour. Eosinophilia (19 % of total leucocytes) was present. Doppler-duplex arterial examination of the upper extremities showed an occlusion of the cubital artery down to the palmar arcade on the right arm. Selective angiography of the right subclavian and brachial arteries showed diffuse alteration of the blood flow in the cubital artery and hand, with fine collateral circulation in the carpal region. Neither secondary causes of hypereosinophilia nor a myeloproliferative process was found. Considering the skin biopsy results and having excluded other causes of eosinophilia, we assumed the diagnosis of an eosinophilic vasculitis. Treatment with tacrolimus and high dose steroids was started, the latter tapered within 12 months and then stopped, but a dramatic flare-up of the vasculitis with Raynaud phenomenon occurred. A new immunosupressive approach with steroids and methotrexate was then introduced. This case of aggressive eosinophilic vasculitis is difficult to classify into the usual forms of vasculitis and constitutes a therapeutic challenge given the resistance to current immunosuppressive regimens.

scholarly journals Corrigendum

1967 ◽  
Vol 63 (4) ◽  
pp. 1031-1031
Keyword(s):  
Second Line ◽  
Right Hand ◽  
The Right ◽  
Range Of Values ◽  
Uniform Interpolation

On Uniform Interpolation SetsBy J. P. EarlUniversity of KentIn the paper referred to in the title (these Proceedings 62 (1966), 721–742) the value of a constant used in the statement of several of the theorems is incorrect. Due to the erroneous appearance of a factor ¼ρ2 in the proof of Theorem 3 (penultimate equation of section 5), the value, or range of values, of τ should in fact be 4/ρ2 times the stated value.There are also some misprints which escaped earlier detection. In the reference of Theorem A d = 0 should read d = D and in the second line of Proof of Theorem 3 Un should be defined by Un ≡ |z − zn| ≤ Δ |zn|1−½ρ. The right-hand side of equation (5.2) should be multiplied by σM, N(z).

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.

scholarly journals Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 404 ◽  
Author(s):  
Hai-Yan Zhang ◽  
Rekha Srivastava ◽  
Huo Tang
Keyword(s):  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Sine Function ◽  
Hankel Determinant ◽  
Third Order ◽  
Toeplitz Determinants ◽  
The Third ◽  
Toeplitz Determinant ◽  
The Right

Let S s * be the class of normalized functions f defined in the open unit disk D = { z : | z | < 1 } such that the quantity z f ′ ( z ) f ( z ) lies in an eight-shaped region in the right-half plane and satisfying the condition z f ′ ( z ) f ( z ) ≺ 1 + sin z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) and Toeplitz determinant T 3 ( 2 ) for this function class S s * associated with sine function and obtain the upper bounds of the determinants H 3 ( 1 ) and T 3 ( 2 ) .

scholarly journals Bilateral Multiple Metacarpal Head Avascular Necrosis: A Case Report

2016 ◽  
Vol 101 (9-10) ◽  
pp. 473-477 ◽  
Author(s):  
Wengang Li ◽  
Biao Liu ◽  
Jun Song ◽  
Yan Liu ◽  
Haoyu Liu ◽  
...  
Keyword(s):  
Avascular Necrosis ◽  
Middle Finger ◽  
Left Hand ◽  
The Third ◽  
Metacarpal Head ◽  
Right Hand ◽  
Satisfactory Clinical Result ◽  
Slight Limitation ◽  
The Right

Avascular necrosis of the metacarpal head is a rare disease. We herein report a case with varying degrees of lesions in the third and fourth metacarpal heads of the right hand and the third metacarpal head of the left hand. The patient was a 37-year-old male right-handed mechanical worker who presented with persistent dull pain in the right hand after labor work for more than a year. The 3 lesions in this patient were treated differently based on their clinical imaging manifestations. The neurologic function of the right hand recovered by the 18-month follow-up; only a slight limitation remained in the right middle finger. This is the first report of 1 patent who received 2 different treatment methods simultaneously and both provided a satisfactory clinical result.

XVIII. Researches on the geometrical properties of elliptic integrals

1852 ◽  
Vol 142 ◽  
pp. 311-416 ◽  
Keyword(s):  
Elliptic Integral ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Mathematical Science ◽  
Third Order ◽  
Geometrical Properties ◽  
First Order ◽  
The Third ◽  
The Right

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.

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.

scholarly journals On solvability of one class of third order differential equations

2021 ◽  
Vol 73 (3) ◽  
pp. 314-328
Author(s):  
B. T. Bilalov ◽  
M. I. Ismailov ◽  
Z. A. Kasumov
Keyword(s):  
Differential Equations ◽  
Fourier Method ◽  
Generalized Solution ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Countable System ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Right Hand ◽  
The Right

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

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