bluuebunny/crossref_metadata_2025 huggingface dataset:
A special net of quadrics
Authors: Edge W. L.
Published: January 2009

It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.
On Vanishing Coaxial Minors
Authors: Motzkin Th
Published: January 2009

In this note we wish to study some properties of a square matrix of order n and of rank r,

whose coaxial minors |M| of certain orders, or some of them, are known to be zero. Our main results will lie in two directions, according as the vanishing minors are, within certain limits, of an arbitrary order (Part II) or of two consecutive orders only (Part III).
On the Equivalence of Two Singular Matrix Pencils
Authors: Williamson J.
Published: January 2009

In a recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.
Some new operational representations
Authors: Humbert P.
Published: January 2009

In this paper I give some operational representations, according to the Carson-Heaviside Calculus, which, as far as I know, are new; and I deduce some properties of the functions thus introduced.
On γ-Matrices and their Application to the Binomial Series
Authors: Vermes P.
Published: January 2009

The known methods of “summing” divergent series, e.g. the means of Cesàro, Riesz, Borel, Lindelöf, Mittag-Leffler are particular cases of the transformation of a sequence (formed from the partial sums) by a T-matrix. An equivalent method is that of the transformation of the series by a γ-matrix, the fundamental properties of which have been proved by Carmichael, Perron and Bosanquet.
On the problem of the electrified disc
Authors: Copson E. T.
Published: January 2009

When a perfectly conducting uniform thin circular disc is kept at a potential V0 in an external electrostatic field of potential Φ, electric charge is induced on the surface of the disc; the problem is to find the surface-density σ of this induced charge and its potential V so that the total potential V + Φ has the constant value V0 on the surface of the disc. This problem was first discussed by Green in 1832, and the solution in the case when there is no external field was deduced by Lord Kelvin from the known formula for the gravitational potential of an elliptic homoeoid. The problem is still of interest since similar ideas occur in the theory of diffraction by a circular disc and in the theory of the generation of sound waves by a vibrating disc when the wave-length is large compared with the radius of the disc.
A question concerning Aronhold's Theorems on Bitangents
Authors: Grace J. H.
Published: January 2009

The question to be discussed is this:—

Can Aronhold's theorems on the bitangents of a quartic curve of genus three be extended to the tritangent planes of a space seztic of genus four?

The answer is “No”: decisiveness arises from the fact that the gist of Aronhold's results can be stated in a very simple form, namely:—

(a) Given seven lines in a plane it is possible to derive from them uniquely and symmetrically a quartic curve that has them for bitangents.
The representation of linear operators on spaces of finitely additive set functions
Authors: Cheney C. A. , De Korvin A.
Published: January 2009

The following changes should be made in the statement of Theorem 3.1 and its proof.

A set function is said to be λ-convex if whenever A and B are in Σ with A ∩ B = φ, then
Remark on a theorem of D. H. Fremlin concerning K-analytic Hausdorff spaces
Authors: Devlin Keith J.
Published: January 2009

At the 29th Meeting of the British Mathematical Colloquium held at Edinburgh in March 1977, D. H. Fremlin announced the following result.

Let Φ denote the proposition that there exists a K-analytic Hausdorff space, having metrizable compacta, which is not Souslin. (We refer the reader to Fremlin's paper (1) for the relevant definitions)

Theorem 1.1. (Fremlin) (i) Assume . (ii) Assume Martin's Axiom together with . Then Φ is false.
Decomposition of a Von Neumann Algebra Relative to a*-Automorphism
Authors: Thaheem A. B.
Published: January 2009

Let X be any real or complex Banach space. If T is a bounded linear operator on X then denote by N(T) the null space of T and by R(T) the range space of T.

Now if X is finite dimensional and N(T) = N(T2) then also R(T) = R(T2). Therefore X admits a direct sum decomposition

.

Indeed it is easy to see that N(T) = N(T2) implies that and, using dimension theory of finite dimensional spaces, that N(T) and R(T) span the whole space (see, for example, (2, pp. 271–73))
Inequalities Involving the Inverses of Positive Definite Matrices
Authors: Merris Russell
Published: January 2009

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined by

For example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.
Tensor products of separable reduced primary modules
Authors: Webb Cary
Published: January 2009

R will denote a Dedekind domain and Pone of its prime ideals. A P-primary module will be an R-module all of whose non-zero elements have annihilators that are powers of the prime P. In all that follows E is such a module.

The height of 0 ≠ x ∈ E will be max{n : x ∈ PnE}. It is denoted by h(x). If this maximum does not exist we will say h(x)∞.

Clearly the condition is equivalent to E having no non-zero elements of infinite height. Adopting the terminology of (2, Ch. XI) where such modules over the ring of integers are studied, we will call these modules separable and reduced.
Joins of Almost Subnormal Subgroups
Authors: Lennox John C.
Published: January 2009

Following (1) we say that a subgroup H of a group G is almost subnormal in G if H is of finite index in some subnormal subgroup of G, or, equivalently, if |Hn : H| is finite for some n, where Hn is the n-th term of the normal closure series of H in G. The aim of this article is to prove, in answer to a question of R. Baer, the following analogue of the well known result of Roseblade and Stonehewer (3) that in any group the join of a pair of finitely generated subnormal subgroups is always subnormal:
The strict topology on a space of vector-valued functions
Authors: Khan Liaqat Ali
Published: January 2009

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.
An Analogue of the Radon-Nikodym Property for Non-Locally Convex Quasi-Banach Spaces
Authors: Kalton N. J.
Published: January 2009

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.

where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach spaces. If 0>p > 1, there are several possible ways of defining “differentiable” operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero.
The partially ordered family of all ℛ-classes of S(x)
Authors: Magill K. D.
Published: January 2009

Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the following

Conjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:

(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.

(2) ℛ(S(X)) and (S{Y)) are order isomorphic.

(3) The semigroupsS(X) and S(Y) are isomorphic.

(4) The spaces X and Y are homeomorphic.
The syntactic near-ring of a linear sequential machine
Authors: Holcombe Michael
Published: January 2009

Let R be a ring, a linear sequential machine over R is a quintuple ℳ = (Q, A, B, F, G) where

Q, A, B are R-modules

F:Q × A →Q and G:Q × A→B are R-homomorphisms.
A singular perturbation problem for the Lavrent'ev-Bitsadze equation
Authors: Weinacht R. J.
Published: January 2009

In this note we consider a singular perturbation problem for the equation

where K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.
The solution of length three equations over groups
Authors: Howie James
Published: January 2009

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.
Unitary bordism of circle actions
Authors: Kosniowski Czes , Yahia Mahgoub
Published: January 2009

The purpose of this paper is to describe , the bordism module of unitary T-manifolds, where T denotes the circle group S1. We give both an algebraic and a geometric description. The algebraic result is

where I = (i(1), i(2),…i(2n)) runs through all finite ordered 2n-tuples (n≧0) of non-negative integers which satisfy the conditions (a) i(l) + i(2n)≠0 and (b) if i(2n)≠0 then i(2n)=≠. The isomorphism is also described geometrically and this leads to geometric generators of .
Nonlinear boundary value problems for elliptic systems
Authors: Cañada A.
Published: January 2009

The purpose of this paper is to discuss non-linear boundary value problems for elliptic systems of the type

where Ak is a second order uniformly elliptic operator and is such that the problem

has a one-dimensional space of solutions that is generated by a non-negative function. The boundary ∂G is supposed to be smooth and the functions gk, 1≦k≦m are defined on Ḡ×Rm and are continuously differentiate (usually, Bk represents Dirichlet or Neumann conditions and is the first eigenvalue associated with Ak and such boundary conditions).
Exit times for elliptic diffusions and BMO
Authors: Bañuelos R. , Øksendal B.
Published: January 2009

In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.
Some examples of minimally degenerate Morse functions
Authors: Kirwan Frances
Published: January 2009

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the form

where Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.
Application of Wynn's epsilon algorithm to periodic continued fractions
Authors: Jamieson M. J.
Published: January 2009

The infinite continued fraction

in which

is periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.
Geometry relevant to the binary quintic
Authors: Edge W. L.
Published: January 2009

The aim in this paper is to indicate one way of interpreting covariants of a binary quintic F geometrically, interpretations having been found recently [7] for its quadratic covariant 2C2, called Γ in [7], and its invariant I4. The symbol dCn will denote a covariant of order n in the binary variables x, y and degree d in the coefficients of F, Id being used in preference to dI0 for invariants. The sum d + n is 4 for both 2C2 and I4, and no other covariant affords as small a sum; so it is natural to have begun by interpreting these two and to use them as auxiliaries in interpreting others.
Easy Geometrical Proof of a Theorem of Chasles
Authors: Press Edward
Published: January 2009

The subjoined is an easy geometrical proof of the following theorem (which derives its importance from being part of the method of Chasles of constructing geometrically the ninth point when eight points of an “associated system” are given).
The Equation x3−u3 = y3−v3 when x, y, u, v are rational
Authors: Steggall J. E. A.
Published: January 2009

Diophantus (Heath, 1885, page 122) is said to have the porism—“The diffeience of two cubes can be expressed as the sum of two cubes.”
On the Linear Differential Equation of the Second Order
Authors: Brodetsky S.
Published: January 2009

The linear differential equation of the second order

is not in general integrable by any method at present available. At the same time, several equations of this type have been integrated, either in terms of finite functions or by means of expansions in series. Some properties of the integrals of the general equation have also been obtained. It is the object of this paper to develop some general properties of these integrals, which throw some light on the nature of the solutions, even if not obtainable in explicit terms.
The Solution of Difference Equations by Continued Fractions
Authors: Strang J. A.
Published: January 2009

The theorems which furnish in C.F. form the roots of a quadratic equation, and the similar process which leads to particular integrals of an ordinary differential equation of the second order, may be applied to certain types of difference equation. The types which suggest themselves for examination are

the bilinear equation, and

a special form of the linear equation; the coefficients are functions of r, and s is constant.
The Co-apolars of a Cubic Curve
Authors: Milne William P.
Published: January 2009

It is a well-known fact and easy to prove that if Γ and Γ′ be any two class-cubics and P a variable point, such that the pencil of lines from P to Γ apolarly separate the pencil of lines from P to Γ′, then the locus of P is a cubic curve G called the “apolar locus” of Γ and Γ′. Also, Γ and Γ′ are said to be co-apolar class-cubics” of G, or simply “co-apolars” of G. The problem of finding the general system of co-apolars of a given cubic curve has not yet been completely solved, but particular and more important cases have been investigated by me in several papers contributed to the London Mathematical Society. In the present communication I propose to deal with the most general solution of the above problem.
A Geometrical Proof of a Theorem of Hurwitz
Authors: Ford Lester R.
Published: January 2009

In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequality

how small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?
Geometrical Illustrations of Cyclant Substitutions
Authors: Taylor D. G.
Published: January 2009

§1. The following three illustrations of cyclant substitutions are supplementary to those given in § 5 of a previous paper.

In the cyclant substitution in three variables with equimodular multipliers

in which the coefficients are real and satisfy the relations

the following expressions, as may at once be verified, are absolute invariants:
On the Failure of Heilermann's Theorem
Authors: Datta Haripada
Published: January 2009

The theorem of Heilermann can be stated thus:—

If the series

is converted into a continued fraction of the form

then the elements of the continued fraction are

where

and is obtained from this determinant by deleting the (r + 1)th column and the last row.
Notes on Euclid I., 47.
Authors: Harvey William
Published: January 2009

ABC is a triangle, right-angled at A, and X, Y, Z are the centres of the squares described on the sides opposite the angles A, B, C; XD, XM, XN are respectively perpendicular to BC, CA, AB; MY, DY are joined, and DY meets AC at E; NZ, DZ are joined, and DZ meets AB at F.
To transform a rectangle into a square
Authors: Peddie W.
Published: January 2009

Let l be the length of the rectangle and b its breadth, while s is the side of the square equal to it in area; s is found, of course, by taking l: s = s : b. A number of different cases arise.
Two Mechanical Integrators or Plammeters
Authors: Fraser A.Y.
Published: January 2009

For the measurement of a plane area, bounded by an irregular curve, various methods are adopted. Besides the well-known methods of approximation in use among land measurers, the following may be mentioned—

The area is divided by parallel ordinates. These are measured, and the results are then treated in several ways to produce more or less accurate results. See Williamson's Integral Calculus, third edition, p. 211.
Kinematical Theorems
Authors: Harvey William
Published: January 2009

§1. Figure 30. Let AB be any straight line in the plane figure, A′B the position of the same line after a displacement, the point A moving to A′ and B to B′. Since the position of a plane figure in its plane is determined, when the position of a straight line rigidly attached to it is determined, the motion of the plane figure is determined if we determine the motion of the straight line AB.
Statical proofs of some Geometrical Theorems
Authors: Alison John
Published: January 2009

1. Figure 54. If ABC be a triangle, P any point, then the system of forces PA, PB, PC is equivalent to the system PH, PK, PL, where H, K, L are the middle points of BC, CA, AB.
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.
On certain forms of Vibration
Authors: Chree Charles
Published: January 2009

Certain cases of motion in a gas contained in a spherical or in a cylindrical envelope or surrounding a sphere or a cylinder have been considered by Professor Stokes and Lord Rayleigh, of which a clear idea may be obtained from chapters xvii. and xviii. of Rayleigh's Sound. Their chief object is to determine the motion in the gas when the motion in the bounding surface is given, or to determine the modification in the nature of a wave being propagated through the gas by the presence of an obstacle.
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, then

1. 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.
On the relations between systems of curves which, together, cut their plane into squares
Authors: Tait
Published: January 2009

If ρ be the vector of a corner of a square in one system, σ that in a system derived without inversion, we must obviously have

k being the unit-vector perpendicular to the common plane.
A short notice of the additions to the Mathematical Theory of Heat since the transmission of Fourier's Memoir of 1811 to the French Academy
Authors: Gibson George A.
Published: January 2009

What is here printed contains merely a list of the memoirs and treatises that may perhaps be found useful for one who wishes to trace the progress of the mathematical theory of heat beyond the stage at which Fourier left it. As discussions of the Fourier series and integrals occur in almost every treatise on the Integral Calculus, I have omitted reference to these. Similar considerations have led me to omit references to the discussion of differential equations, except where these specially dealt with the problem of the conduction of heat.
On the solution of the equation xp–1=0 (p being a prime number)
Authors: Butters J. Watt
Published: January 2009

[At the first meeting of this Session a paper was read on the value of cos 2π/17, which evidently may be made to depend on the solution of x17 – 1 = 0.* The present paper is the outcome of a suggestion then made, that a sketch of Gauss's treatment of the general equation might prove interesting. To give completeness to the subject the necessary theorems on congruences have been prefixed. The convenient notation introduced by Gauss is here adopted; thus, when the difference between a and b is divisible by p, instead of writing a = Mp + b, we may write a ≡ b (mod p), the value of M seldom being of importance. It is evident that if a ≡ b, then na ≡ nb, and an ≡ bn, n being any positive integer, and the same modulus p being understood throughout. Also a/n ≡ b/n provided n be prime to p. Other properties (similar to those of equations) are easily seen, but only the above are needed here.
Note on the regular solids
Authors: Steggall
Published: January 2009

The usual methods of proving the existence of regular polyhedra, as given in Wilson and in Todhunter, appear to most students somewhat difficult. It seemed worth while trying, therefore, whether a simpler or more direct proof could not be obtained. The following note shows how this may be done.
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 is

incidentally 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.
Sur un Lieu Géométrique
Authors: Aubert M. Paul
Published: January 2009

Par un point fixe A d'une circonférence donnée on mène deux cordes AB et AC dont le produit a une valeur constante m2, puis on joint BC. Trouver 1° le lieu du pied D de la bissectrice de l'angle A du triangle ABC; 2° le lieu des centres des cercles inscrits et exinscrits à ce triangle.
The triangle and its escribed parabolas
Authors: Pressland A. J.
Published: January 2009

In what follows, a triangle ABC is taken, from it another triangle A′B′C′ is formed so that

A is the mid point of B′C′,

B is the mid point of C′A′,

and C is the mid point of A′B′.
On some properties of a triangle of given shape inscribed in a given triangle
Authors: Allardice R. E.
Published: January 2009

It is well known that in a given triangle a one-fold infinity of triangles may be inscribed similar to a given triangle This becomes at once obvious on consideration of the converse problem; for we may circumscribe about a given triangle (A), a triangle similar to a second triangle (B), and having its sides parallel to the sides of (B).
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 transformation and classification of Permutations
Authors: Sprague T. B.
Published: January 2009

Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations
On the numerical values of the roots of the equation cosx = x
Authors: Miller T. H.
Published: January 2009

The angles which satisfy the equation

x=cosx,

occur in the solution of certain problems in analytical geometry.

It is easy to show that one value of the angle is coscoscos…cos 1, when the cosine is taken successively to infinity.
Note on an Equation of Motion
Authors: Pressland A. J.
Published: January 2009

It can be shown by means of relative motion that if two bodies A and B move with velocities u and v in the same straight line, and a third body C move with velocity u + v also in the same straight line, the space passed over by C is equal to the sum of the spaces passed over by A and by B in the same time.
On the smallest number of entries necessary in a table of logarithms to seven decimal places
Authors: Steggall J.E.A.
Published: January 2009

Taking seven figure logarithms it is required to find at what interval from n, a number of five figures, the next entry need be made, in order that any intermediate logarithm may be calculated by the method of proportional parts.

We have
Deduction of the Thermodynamical Relations
Authors: Peddie William
Published: January 2009

In his Theory of Heat, Clerk-Maxwell gives a very elegant and simple geometrical proof of the four thennodynamical relations, and points out that his construction shows that the truth of any one is a necessary consequence of the truth of any other. The usual analytical proof of these relations can be made as simple as Maxwell's geometrical proof, and the fact that any one is a necessary consequence of any other becomes evident when it is considered that they are all deduced by a common process from identical transformations of one equation. The following method of proof seems to me to bring out more directly their necessary interdependence.
On Lines of Magnetic Force
Authors: Steggall J.E.A.
Published: January 2009

In my laboratory we make great use of the method of tracing lines of force described in Glazebrook and Shaw's Practical Physics. The field is modified in various ways; for instance, the fixed magnets are sometimes placed so as to somewhat resemble in their disposition the field magnets of a dynamo: sometimes a circular piece of soft iron is placed in the field, and the effect of its induced magnetization examined. The students like these exercises, and the results (of which some are exhibited) are very beautiful. We use steel bars about 8 cm. long for fixed magnets, and small rather heavy lozenge-shaped needles, about one to two cm. long, as what I shall call pointers.
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.

Let

denote 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.
Theorems connected with Simon's Line
Authors: Burgess A. G.
Published: January 2009

If XYZ is the Simson Line P(ABC); if PM is perpendicular to XYZ and cuts the sides BC, CA, AB in U, V, W: then PU.PV.PW=PA.PB.PC=4R2.PM.
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.
A Method for Successive Graphic Integrations
Authors: Muirhead R. F.
Published: January 2009

The method here described applies to the successive integrations of any function whose graph consists of segments of straight lines.
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, although

is one of an orthogonal triad, φ does not satisfy Cayley's equation.
An Extension of Clairaut's Differential Equation
Authors: Levy H.
Published: January 2009

I propose in this paper to investigate the geometrical nature of a certain species of differential equation, which includes, as a particular case, Clairaut's well worn equation
On Graphical Integration
Authors: Goldziher Charles
Published: January 2009

The theory of graphical integration is founded on the graphical integration of parabolic arcs. The details and the different technical applications of these problems are developed in the fundamental work of J. Massau; we find there very simple pure constructive methods for the case of parabolas till the 3rd order.
Proof of a Fundamental Relation in the Theory of Bending between the Bending Moment and Load Curves, or between the Deflection and Bending Moment Curves
Authors: Horsburgh E. M.
Published: January 2009

The important relation in the Theory of Bending between the curves of Bending Moment (B.M.), Shearing Force (S.F.), and Load, or between those of Deflection, Slope, and Bending Moment, viz., that the tangents to the first of either set intersect in a vertical line through the centroid of the corresponding area of the last, under the usual convention of drawing, is usually not proved in Engineering Treatises, or else is established in simple cases by the polygon of loads.
The Fundamental Formula for the Area of a Triangle in Analytical Geometry
Authors: Tweedie C.
Published: January 2009

The algebraic methods employed are usually defective in explaining the sign to be attached to an area; while the trigonometrical method with the transformation of a trigonometrical formula has always seemed a little far fetched so early in analytical geometry.
A mechanism for solving equations of the nth degree
Authors: Muirhead R. F.
Published: January 2009

The proposed mechanism explained here is based on the geometrical properties of Fig. 1.

0ABO′ is a straight line having 0A = BO′ = 1, and OB = AO′ = x and AH, BK are lines perpendicular to AB.
A Problem in the Linear Flow of Heat discussed from the point of view of the Theory of Integral Equations
Authors: Carslaw H. S.
Published: January 2009

In the first section of Kneser's book on Integral Equations and their Applications to Mathematical Physics, he applies that theory to the solution of some of the problems which arise in the Linear Flow of Heat. The object of this paper is to illustrate Kneser's use of Integral Equations in the Mathematical Theory of the Conduction of Heat by the discussion of one of the classical problems of Linear Flow which he leaves untouched.
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 determinant

each 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 follows

where 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 Simple Linkage for Describing Equal Areas
Authors: Horsburgh E. M.
Published: January 2009

If H (Fig. 1) be the middle point of a straight bar QP and if a straight bar OH of length one-half of QP be pin-jointed to QP at H, a simple linkage is formed, which may be called a T -linkage.
The Solutions of Mathieu's Differential Equation: Representation by Contour Integrals, and Asymptotic Expansions
Authors: Dougall John
Published: January 2009

It is an obvious remark that the Mathieu functions, being the harmonic functions of the elliptic cylinder, must be closely related to the Bessel functions, the harmonic functions of the circular cylinder. Reference has been made to some aspects of this relationship in two earlier communications, to which the present paper may be regarded as a sequel.
Einsteinian Gravitational Field of a Heterogeneous Fluid Sphere
Authors: Ghosh Jyotirmaya
Published: January 2009

The field-equations of gravitation in Einstein's theory have been solved in the case of an empty space, giving rise to de Sitter's spherical world. In the case of homogeneous matter filling all space, the solution gives Einstein's cylindrical world. The field corresponding to an isolated particle has been obtained by Schwarzchild. He has also obtained a solution for a fluid sphere with uniform density, a problem treated also by Nordström and de Donder. A new solution of the gravitational equations has been obtained in this paper, which corresponds to the field of a heterogeneous fluid sphere, the density at any point being a certain function of the distance of the point from the centre. The law of density is quite simple and such as to give finite density at the centre and gradually diminishing values as the distance from the centre increases, as might be expected of a natural sphere of fluid of large radius. The general problem of the fluid sphere with any arbitrary law of density cannot be solved in exact terms. It will be seen, however, from a theorem obtained in this paper, that the solution depends on a linear differential equation of the second order with variable coefficients involving the density, and thus the laws of density for which the problem admits of exact solution are those for which the above coefficients satisfy the conditions of integrability of the differential equation. An approximate solution for any law of density may be obtained by the method of series.
A Note upon some Properties of the Curve of Striction
Authors: Richmond H. W.
Published: January 2009

Certain properties of Ruled Surfaces relating to their Curves of Striction are suggested and immediately proved by the use of a particular kind of coordinates, known as Dual Coordinates, for the Generating Lines. These coordinates, were introduced by Prof. E. Study; the theory of them is fully explained and many beautiful applications are made in his treatise, Geometrie der Dynamen, Teubner, 1903, and in his article, Complexe Grössen in the Encyclopädie der math. Wissenschaften, Bd. I., pp. 147–183, and specially p. 166. In applying the method here I have ventured to regard the fundamental quantity c in a light which, if less rigorous, has the advantage of being familiar to most of us in other mathematical work.
Applications of Mathematics to Medical Problems
Authors: M'Kendrick A. G.
Published: January 2009

In the majority of the processes with which one is concerned in the study of the medical sciences, one has to deal with assemblages of individuals, be they living or be they dead, which become affected according to some characteristic. They may meet and exchange ideas, the meeting may result in the transference of some infectious disease, and so forth. The life of each individual consists of a train of such incidents, one following the other. From another point of view each member of the human community consists of an assemblage of cells. These cells react and interact amongst each other, and each individual lives a life which may be again considered as a succession of events, one following the other. If one thinks of these individuals, be they human beings or be they cells, as moving in all sorts of dimensions, reversibly or irreversibly, continuously or discontinuously, by unit stages or per saltum, then the method of their movement becomes a study in kinetics, and can be approached by the methods ordinarily adopted in the study of such systems.
The Anharmonic Ratios of 4 points in a straight line
Authors: Muirhead R. F.
Published: January 2009

The anharmonic ratio of four points A, B, C, D depends on the order in which they are taken. The A.I.G.T. “Syllabus,” following Cremona, defines the symbol (ABCD) to stand for , when A is conjugate to B and C to D. Thus, taking λ to denote the value of this double-ratio, we have
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.
The locus of points at which two sides of a given triangle subtend equal or supplementary angles
Authors: Naraniengar M. T.
Published: January 2009

1. Let the sides AB, AC subtend equal angles λ at P.

Take ABC as the triangle of reference and denote AP, BP, CP by x, y, z. Then we have, using trilinears,
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 form

m1, 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 relation

where n is a positive integer. He showed that they satisfied the differential equation

that they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form