bluuebunny/crossref_metadata_2025_split huggingface dataset:
Note on Euclid II. 11.
Authors: Childers Hon Hugh C. E. 
Published: January 2009

[The foflowing method of dividing a straight line in medial section was communicated to Mr Munn of the Edinburgh High School, by the Right Hon. Hugh C. E. Childers in January last.]Let AB be the line. (Fig 56).Draw AC = ½AB, and at right angles to it. Join CB. Bisect the angle ACB with CD, cutting AB at D. Draw DE at right angles to CB. Then the triangles CAD, CED are equal, and AD = DE. Draw a circle with DA and DE radii, cutting AB at F.
A Problem in Combinations
Authors: Robertson Alexander 
Published: January 2009

Given sets of balls of different colours, in how many ways may they be arranged in line so that no two balls of the same colour shall come together.If we have two colours only, and the same number ‘m of each colour, there are evidently two arrangements possible; if we have m, m – 1 respectively, only one arrangement is possible; if we have m, m – 2 ; in, m – 3, &c., no arrangement is possible. We may write these results
Trigonometrical Mnemonic
Authors: Alison John 
Published: January 2009

If the functions be written as in the above figure, then1. Of any three functions written consecutively on a circular arc or on a diameter, the mean is the product of the extremes. Thus sinA = tanAcosA, 1 = sinAcosecA, and so on.2. In each triangle the square of the function written at the vertex turned downwards is equal to the sum of the squares of the functions written at the other two vertices. Thus I2 = sin2A + cos2A, and so on.
A certain cubic connected with the triangle
Authors: Steggall  
Published: January 2009

In examining some of the lines that occur in connection with the recent geometry of the triangle, the cubic whose equation in trilinear co-ordinates isincidentally appeared, and it seems worth noting the very large number of special points it passes through. These are the vertices, the mid-points of the sides, the inscribed and escribed centres, the circumcentre, the orthocentre, the centroid and the symmedian points, or fourteen in all.
On some applications of the pedal line of a triangle
Authors: Anglin A. H. 
Published: January 2009

1. Taking the two following known properties of the pedal line of a triangle, viz.:I. The locus of a point, such that the feet of the perpendiculars from, it on the sides of a triangle are collinear, is the circum-circle of the triangle;II. The pedal line bisects the distance between the orthocentre and the corresponding point in the circumference of the circum-circle;
On the relation between the stereographic projections of points of a plane related to one another by inversion
Authors: Tweedie Charles 
Published: January 2009

What is meant will best be understood by the following:-1. Suppose we have an unlimited straight line XX′ (fig.24) with a finite point O on it. Take two points on the line, C and C1 in the same sense with respect to O such that OC·OC1=a2. Then C and C1 are connected by inversion, and O is the centre inversion. Without loss of generality for what follows we may, for convenience, take a2=12=1, so that OC·OC1=1.
On the number of Independent Conditions involved in the vanishing of a Rectangular Array
Authors: Sommerville D. M. Y. 
Published: January 2009

1. The notation for a rectangular array can be extended so as to admit of arrays in which the number of rows exceeds the number of columns.Letdenote the aggregate of all determinants of the mth order which can be formed from the rectangular array of pq elements by deleting p – m columns and q – m rows.
A proof that the middle points of parallel chords of a conic lie on a fixed straight line
Authors: Jack  
Published: January 2009

Let S be the focus of the conic, PX the directrix, e the eccentricity.Let V be the middle point of PP′.Draw VK perpendicular to the directrix, and with centre V describe a circle, radius equal to e.VK.Join SP, SP′ and draw radii Vp, Vp′, parallel to SP′, SP, and let PP′ meet the directrix in L. Then p, p′ are on the line SL.
1. A proof of Dupin's theorem with some simple illustrations of the method employed. 2. Two methods of obtaining Cayley's condition that a family of surfaces may form one of an orthogonal triad. 3. An extension of Dupin's theorem to the case in which a family of surfaces is cut orthogonally by two other families which intersect at a constant angle, with the condition that a family may be capable of being cut in this manner
Authors: Edwardes F. E. 
Published: January 2009

A proof of Dupin's theorem with some simple illustrations of the method employed.Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.
The condition that the family of surfaces φ(x, y, z, λ) = 0 may be one of an orthogonal triad
Authors: Edwardes F. E. 
Published: January 2009

Cayley has shown that if the family of surfaces φ(x, y, z) = λ is one of an orthogonal triad, φ satisfies a differential equation of the third order. If, however, the parameter λ is involved implicitly in the equation of the family, the condition requires modification: for example, althoughis one of an orthogonal triad, φ does not satisfy Cayley's equation.
Linear Substitutions and their Invariants
Authors: Taylor D. G. 
Published: January 2009

The main points of this paper are :—(i) The construction of a linear substitution from its poles or linear invariants and its multipliers (§§3, 7);(ii) a formula for r repetitions of a substitution (§ 4);(iii) a specification of the types of linear substitutions of order r, with examples of the simplest of those types(§§9ff.);(iv) The geometrical illustration of the case of three variables (§§15 ff.)
On a certain class of Linear Substitutions with Common Invariants, and an Associated Substitution of Order Four
Authors: Taylor D. G. 
Published: January 2009

The determinanteach row of wliich contains the same n elements in the same cyclic order, with ′a1 always in the leading diagonal, is the product of n linear factors, which we shall write as followswhere ρ is any primitive nth root of unity.
A Problem of Robert Simson's
Authors: Philip G. 
Published: January 2009

The following problem appears in Robert Simson's “Opera Quaedam Reliqua,” pp. 472–504 :“ Si a duobus punctis datis A, B ad circulum positione datum CDE inflectantur utcumque duae rectae AC, BC circumferentiae rursus in D, E occurrentes; juncta DE vel continebit datum angulum cum recta ad datum punctum vergente ; vel parallela erit rectae positionae datae; vel verget ad datum punctum:” i.e. if from two given points A and B any two straight lines AC, BC are drawn to a circle CDE given in position, and they meet the circumference again in D and E, then the straight line DE (I.) will inake a constant angle with a straight line passing through a fixed point, or (II.) will be parallel to a straight line given in position, or (III.) will pass through a given point. This final form of the result was only arrived at by Simson after he obtained the aid of Matthew Stewart.
A Problem in Voting
Authors: Sommerville D. M. Y. 
Published: January 2009

The following is a problem in voting:A number of n individuals are separately arranged in order of preference by a number of k electors ; it is desired to ascertain the order of preference in the opinion of the electors as a whole.Without discussing what is the best solution of this problem, there are three plausible methods, and it will appear as the result of a curious theorem that two of them lead to identical results.
Gauss's Theorem on the Regular Polygons which can be constructed by Euclid's Method
Authors: Carslaw H.S. 
Published: January 2009

The methods now adopted in the teaching of elementary geometry have made it most important that the teacher should have clear views upon the nature of the problems which are soluble by Euclid's methods: that is, with the aid of the ruler and compass only. With this general question I have dealt in another place. In this paper I give a short account of the argument by means of which Gauss proved that the only regular polygons of n sides, which can be constructed by Euclid's methods, are those in which n, when broken up into prime factors, takes the formm1, m2, m3,…mr being all different.
Triangles Triply in Perspective
Authors: M‘Leod Charles , Milne William P. 
Published: January 2009

The following way of viewing triangles triply in perspective is of interest.Two triangles ABC and PQR are said to be “in perspective” if the joins of corresponding vertices be concurrent.Thus if U be the point of concurrency we have the following collineations:—2. Triangles doubly in perspective are also triply in perspective.Let the two triangles ABC and PQR be doubly in perspective, the two centres of perspective being U and V.
Discussion and History of Certain Geometrical Problems of Heraclitus and Apollonius
Authors: Archibald R. C. 
Published: January 2009

Not the least interesting portions of the wonderful “Mathematical Collections” of Pappus are those which reproduce parts of the νε⋯σεις, or the two lost books of Apollonius (247–205 B.C.). Pappus (c. 300 A.D.) writes:— “A line is said to verge (using Heath's translation) toward a point if, being produced, it reach the point,” and among other particular cases of the general problem he gives the following as treated by Apollonius:Problem A: Between two lines, given in position, to place a straight line given in length and verging toward a given point.
On the Equation of the Parabolic Cylinder Functions
Authors: Milne Arch 
Published: January 2009

Hermite, in 1864 (Comptes Rendus, vol. 58) introduced into analysis the polynomials defined by the relationwhere n is a positive integer. He showed that they satisfied the differential equationthat they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form
On some results concerning integral equations
Authors: Humbert Pierre 
Published: January 2009

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.
Notes on Spherical Harmonics
Authors: Dougall John 
Published: January 2009

This paper contains(a) an explicit solution of the problem of finding a function which is harmonic within a given sphere and takes at the surface the same value as a given rational integral homogeneous function of the rectangular coordinates of a point referred to the centre of the sphere as origin;(b) a concise symbolical expression for the integral, over the surface of the sphere, of the product of any three rational integral spherical harmonics.
On the general solution of Mathieu's equation
Authors: Whittaker E. T. 
Published: January 2009

The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the typewhere A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.
On the Quasi-periodic Solutions of Mathieu's Differential Equation
Authors: Young Andrew W. 
Published: January 2009

Mathieu's differential equationis the equation which arises out of those two-dimensional problems Mathematical Physics in which the boundary is an ellipse, such problems, for example, as the vibrations of an elliptic membrane, which was first discussed by Mathieu,* and the scattering electromagnetic waves by a wire of elliptic cross-section. A different use of the same equation is found in Celestial Mechanics the treatment of perturbations and oscillations about periodic orbits, and, in a more mundane connection, it has been shown to be the differential equation of the variety artiste who holds an assistant poised on a pole above his head while he himself is standing on a spherical ball rolling on the ground!
The six trisectors of each of the angles of a triangle
Authors: Taylor F. Glanville , Marr W. L. 
Published: January 2009

1. The following is an account of a theorem whose origin has been traced to Prof. Morley of Johns Hopkins University.In the course of certain vector analysis, some 14 years ago, Prof. Morley found that if a variable cardioide touch the sides of a triangle the locus of its centre, that is, the centre of the circle on which the equal circle rolls, is a set of 9 lines which are three by three parallel, the directions being those of the sides of an equilateral triangle. The meets of these lines correspond to double tangents; they are also the meets of certain pairs of trisectors of the angles, internal and external, of the first triangle. This result was never published, and it was only the particular case of the internal trisectors that reached the present writers, the existence of the enveloping cardioides and the set of 9 lines being quite unknown to them.
Morley's Trisection Theorem: an Extension and its Relation to the Circles of Apollonius
Authors: Marr W. L. 
Published: January 2009

In Morley's trisection theorem there are three triads of parallel lines which by their intersection with each other form equilateral triangles. The three lines EF, E11F11, E22F22 (v. Taylor and Marr) form one of these triads, and the equations are:—whereand β γ β1, γ1, etc., are similarly defined.
On the plane representation of the homaloidal surfaces which have a twisted cubic as multiple curve
Authors: Tinto J. F. 
Published: January 2009

The most important application of the rational point transformations between two spaces lies in the construction of algebraical surfaces possessing singularities of various kinds and the investigation of their properties.When a plane is transformed by such a transformation into a rational algebraical surface (homaloid) the geometry of the surface is immediately derivable from the geometry of the plane.
On a Group of Transformations connected with the 27 Lines of the Non-Singular Cubic Surface
Authors: Tinto J. F. 
Published: January 2009

In the paper, “Transformations founded on the Twisted Cubic and its Chord System,” a series of space transformations was described, each of which had the property of transforming the chord system of one cubic into the chord system of the other. In the present paper it is shown that by the aid of a non-singular cubic surface the transformations of orders 1, 2, 3, 4, 5 may be derived directly without the intervention of a space transformation.It will be found that these transformations form a group which is intimately associated with the 27 lines of the cubic.
The Brocard and Tucker Circles of a Cyclic Quadrilateral
Authors: Brown Frederick G. W. 
Published: January 2009

The application of the geometrical properties of the Brocard and Tucker circles of a triangle to a quadrilateral appears never to have been adequately worked out, as far as the author can discover. Hence, the object of this paper.Some of the problems involved have been published, under the author's name, as independent questions for solution, and where, in the author's opinion, solutions other than his own have seemed more satisfactory for the logical treatment of the subject, these solutions have been employed, with due acknowledgments to their authors.
Rolling Loads: a New Graphical Method
Authors: Muirhead R. F. 
Published: January 2009

For determining the greatest bending moment produced by a rolling load consisting of two concentrated loads at a fixed distance from one another passing over a span, the usual method involves somewhat intricate algebraic discussion. Even for two concentrated loads the discussion is complicated, and for three it would be still more so.The method here explained, which applies directly to the case when the whole of the rolling load is on the span, uses the shearing force curve only, and gives a very simple geometrical construction for finding the position and magnitude of the greatest B.M.We will begin by stating the construction, and then proceed to give a geometrical proof.
An Analytical Treatment of the Cam Problem
Authors: Stokes G. D. C. 
Published: January 2009

The cam problem may be stated as follows:To design a curve rotating about O so as to actuate by contact a given follower curve to rotate about O′ in a prescribed periodic manner.Reference may be made to Barr, Kinematics of Machinery, Chapter 5, for the engineer's method of constructing cams by means of a template. A fuller analysis of the problem is aimed at in this paper.
The Vibrations of a Particle about a Position of Equilibrium
Authors: Baker Bevan B. , Ross E. B. 
Published: January 2009

1. In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. Considering the oscillations of a dynamical system having a number of degrees of freedom, and supposing to be the frequencies corresponding to infinitesimal oscillations in the different normal coordinates, Korteweg has shown that these cases of interest arise only whenis zero or very small, where p1, p2,… are small integers, positive or negative; the most important cases occur when
On the Riemann P Equation with Integral Exponent differences
Authors: Ferrar W. L. 
Published: January 2009

The differential equation in question is of the second order and has three regular singular points. It is usually denoted bywhere a, b, c are the singular points, a, a′ etc. the exponents at those singularities. The solution when no one of the numbers α – α′, β – β′, γ – γ′ is an integer or zero is well known; all types of solution are expressible in terms of the hypergeometric functions.
Note on the paper, of Dr Bevan B. Baker, An Extension of Heaviside's Operational Method of Solving Differential Equations
Authors: Kaucký T. 
Published: January 2009

Let f(x) and F(x) be polynomials which are supposed to be decomposed into a series of n terms asFurther let θ be a distributive operation and φ(x) be a given function. Then the functional equationhas a solution of the formwhere yν (x) (ν = 1, 2, … n) is the solution of the equation
The Properties of a new Orthogonal Function Associated with the Confluent Hypergeometric Function
Authors: Chappell George E. 
Published: January 2009

The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equationand can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,
Frontal recess cells: a rare cause of frontal sinus pneumocoele
Authors: Dixon J F , Woodworth B A 
Published: January 2009

AbstractBackground and aims:A pneumocoele is a pathologically expanding, air-containing paranasal sinus. Pneumocoeles are uncommon entities, with very few cases reported in the literature. The most plausible aetiological theory is that of a one-way valve effect allowing air into the sinus under increased pressure without pressure equilibration.Methods:Review of a frontal sinus pneumocoele caused by a type III frontal cell and intersinus septal cell.Results:We present a patient with chronic sinusitis who developed a pneumocoele of her left frontal sinus with erosion into her orbit. This patient was a habitual, chronic nose-blower with a large type III frontal cell adjacent to an intersinus septal cell.Conclusion:We propose the new theory that, in our patient, such cells formed a one-way valve, causing air-trapping and pathological expansion of the frontal sinus. To our knowledge, there are no previously published cases that establish an association between large frontal cells and expansion of a pneumocoele.
Suturing of Little's area of the nasal septum for epistaxis
Authors: ZhengHua Z , Gang F , BingWei Z , JiaWen C 
Published: January 2009

AbstractBackground:Epistaxis is a common condition presenting to emergency and ENT clinics. Most epistaxis bleeding originates from Little's area of the nasal septum. Some cases of refractory epistaxis cannot be resolved with packing or electrocoagulation.Method:The suturing technique applied is widely used in surgery. When faced with pulsatile, recurrent bleeding from Little's area of the nasal septum, all rhinologists should possess the skills to be able to suture the bleeding site, especially after failure of local cautery or packing.Case report:The authors have successfully treated four difficult cases of nasal bleeding from Little's area by suturing.Conclusion:This technique could eliminate the discomfort and complications of nasal packing. To the best of our knowledge, this report probably represents the first published case series of suturing of Little's area to treat recurrent nasal bleeding. More cases need to be treated in this way to enable further evaluation of the technique.