bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Summability of Series by a Method of Valiron
Authors: Hyslop J. M.
Published: January 2009

The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s if

This is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.
The asymptotic periods of integral and meromorphic functions
Authors: Whittaker J. M.
Published: January 2009

In a former paper published in these Proceedings it was shown that an integral function of order less than 1 cannot have any asymptotic periods, and it was suggested that a function of order 1 can have at most a set Kω(k=±1, ±2, …). This was subsequently found to be the case. Meromorphic functions for which K, the exponent of convergence of the poles, is less than ρ, the order, behave in many ways like integral functions, so we should expect that (i) if 0≦κ<ρ1 there should be no asymptotic periods, (ii) 0≦κ<ρ1 either none or else a single sequence kω(k = ±1, ±2, …). It will be shown that this is so.
A Theorem on Alternants
Authors: Littlewood D. E.
Published: January 2009

The following is a direct proof of a theorem by Zia-ud-Din.

Let {ν} ≡ {ν1, ν2, …, νp)} be any S-function of weight r + s such that

in the alternant denoted in the theorem by A(αβγ….). Let {μ} be an S-function of weight s, equal to and let {λ} be an S-function of weight r, such that {λ}δ(x1, …,xp) is obtained with coefficient gλμν by diminishing the indices in the alternant {ν}δ(x1, …,xp) according to the theorem.
Asymptotic representation of Laplace transforms with an application to inverse factorial series
Authors: Erdélyi A.
Published: January 2009

In a recent paper Professor A. C. Aitken directed attention to a little known type of inverse factorial series, which he calls inverse central factorial series. He showed that this type of series is a very powerful means of asymptotic representation of functions of a certain type, and concluded his note with the-remark that there is here scope for considerable investigation. As a modest contribution to this investigation, the asymptotic theory, foreshadowed in the last paragraph of Aitken's paper, will be given here: it is hoped to give the convergence theory of inverse central factorial series later.
On Certain Discontinuous Wave Functions
Authors: Erdélyi A.
Published: January 2009

Among the many solutions of the wave equation investigated by the late Professor Bateman there is one type which has so far received little attention.
Two Multiplier Theorems for H1(U2)
Authors: Oberlin Daniel M.
Published: January 2009

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp if

This paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1
Functorial radicals and non-abelian torsion, II
Authors: Feigelstock Shalom , Klein Aaron
Published: January 2009

The object of this paper is to complete and continue some matters in [1].

In [1], Section 2, the torsion and torsion-free functors, whose operation on the category of abelian groups are well known, were extended to the category of all groups as follows. For a group A, put t0(A)= 0 and t1(A) = the subgroup of A generated by the torsion elements of A. Inductively define tn+1(A)/tn(A)=t1(A)/tn(A)), for every positive integer n. Then T(A)=∪ntn(A) is the smallest subgroup H of A such that A/H is torsion-free, [1], Th. 2.2. A group A satisfying T(A) = A was called a pre-torsion group. In [1], 2.12 an example was constructed of a group A satisfying t1(A)≠t2(A)=A. The question was posed whether for every positive integer n there exist groups A, satisfying tn–1(A)≠tn(A)=A. Here we give an affirmative answer. In fact, such groups will be constructed, as well as pre-torsion groups A with tk(A)≠A for every positive integer k, see Section 1.
Isomorphisms between spaces of vector-valued continuous functions
Authors: Kalton N. J.
Published: January 2009

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.
Unitary representations corresponding to measures with bounded support in infinite dimensions
Authors: Galé José E.
Published: January 2009

Let E be a real Hausdorff locally convex space with topological dual E′, topologised by the strong topology. Let (x, x′) denote the bilinear mapping defining the duality between E and E′ (x∈E, x′∈E′). By a unitary representation of E′ we mean an operator valued function U(x′) = Ux′. defined on E′, whose values are unitary operators in a separable Hilbert space H such that
On the automorphisms of the group ring of a unique product group
Authors: Mehrvarz A. A. , Wallace D. A. R.
Published: January 2009

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.
On the centre and residual finiteness of the automorphism group of a group ring
Authors: Wallace D. A. R.
Published: January 2009

Let G be a group and let Aut(G) be its automorphism group. It is notorious that the properties of Aut (G) do not relate well to the properties of G, perhaps the only twogeneral results being that if G has a trivial centre then the same is true of Aut (G) [2, p.89] and Baumslag's theorem that if G is finitely generated and residually finite then Aut (G) is also residually finite [1, Theorem 1, p. 117]. In the paper we shall attempt tofind analogues of these results for therelationship between the properties of R(G), the group ring of G over a ring R, and the properties of Aut R(G), the automorphism of R(G). We prove that if R(G) has a trivial centre then Aut R(G) has a trivial centre. We establish the analogue, Theorem 2.3, of Baumslag's theorem by ring-theoretic methods; our original proof used properties of group rings, the present simplified proof we owe to the referee. As an example we calculate Aut ℤ(G) in the case that G is the direct product of two cyclic groups, one of infinite order and the other of order 5. This calculation will, it is hoped, give some indication of the difficulties in determining automorphisms of the group ring of an infinite group.
The space of totally bounded analytic functions
Authors: Horwitz Alan L. , Rubel Lee A.
Published: January 2009

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.
Geometrical aspects of the system and applications to the nonlinear wave equation
Authors: Cieciura G. , Grundland A. M.
Published: January 2009

Let E be n-dimensional (n≧2) real vector space with a nondegenerate symmetric scalar product (.|.):E × E → R1 with an arbitrary signature (p, n–p). Let us consider a second order partial differential equation (P.D.E.) of the form:

where φ is a given function of two variables, v is an unknown function (defined on an open subset 0 ⊂E), |∇ν|2: =(∇ν|∇ν) is the square of the gradient ∇ν of the function ν and ∇2, denotes the Laplace-Beltrami operator.
Corrigenda: Inverse multi-parameter eigenvalue problems for matrices II
Authors: Browne Patrick J. , Sleeman B. D.
Published: January 2009

In the paper referred to above (see [1]), the following corrections should be made.

(1) Hypothesis 2.2 should read:

(2) The set E⊂ℝn should be defined as

(3) F0(υ)=υ+x should read F(υ)=υ–x.

(4) The unique solution of F0(υ)=0 is υ=x (not υ= –x as printed).

(5) Page 345, line 14:
Fourier's Integral
Authors: Brown T. A.
Published: January 2009

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.
On the Latent Roots of Compound Determinants and Brill's Determinants
Authors: Whittaker E. T.
Published: January 2009

The roots of the equation in λ

were named by Sylvester the latent roots of the determinant | a11a22 … ann |. So early as 1852, Sylvester showed that if any determinant D is given, we can at once write down a determinant whose latent roots are the squares of the latent roots of D: this determinant is in fact the square of D, the process of squaring being performed by multiplying rows into columns : so that, e.g. if the latent roots of are λ1 and λ2, then the roots of are and . Spottiswoode had also shown in 1851 that the latent roots of the reciprocal of a determinant are the reciprocals of the latent roots of the determinant itself. Both these theorems were soon found to be particular cases of a general theorem which was enunciated by Sylvester thus: The latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself.
The Apolar Locus of Two Tetrads of Points
Authors: Milne William P.
Published: January 2009

In the present paper I propose to investigate the fundamental geometrical properties of the Apolar Locus of two tetrads of points in a plane.

The Apolar Locus of two tetrads of points K, L, M, N and P, Q, R, S is defined in the locus of the point X, moving so that the pencils X[K, L, M, N] and X[P, Q, R, S] are apolar.
On a Class of Continued Fractions
Authors: Ford Lester R.
Published: January 2009

The continued fractions treated in this paper are of the general form

where s0, s1, s2, … are real integers (positive, negative, or zero). An arbitrary real number can, of course, be developed in such a fraction in an infinite variety of ways. The continued fractions discussed here have a number of striking properties and present numerous contrasts with the ordinary continued fraction usually employed.
On the Theory of Continued Fractions
Authors: Datta Haripada
Published: January 2009

The present paper is a further contribution towards the object defined in my former paper, namely, to derive the principal known results regarding Continued Fractions, and some new theorems, by transforming the functions considered from infinite series to Continued Fractions, by use of the theory of determinants.
On the Computation of a Lagrangian Interpolation
Authors: Young A. W.
Published: January 2009

Interpolation is one of the most frequent processes in calculation, and yet it is the process in which most computers find the ordinary methods least satisfactory and most troublesome. Indeed, whenever linear interpolation is not practicable, it is usually worth while to find out a method depending on the nature the functions involved in the calculation, and use it in preference to the ordinary difference or Lagrangian formulae. In interpolation by differences there is the want of adequate tables of the coefficients, and worse than that, the necessity for watching the signs and the decimal points, a necessity which in these days calculating machines is relatively a great trouble. There is usually, moreover, a lack of system about interpolation by differences that makes it peculiarly susceptible to slips of working. In this connection I might mention a useful and not too well-known arrangement of the work for Newton's formula which Legendre gives in his Traité des fonctions elliptiqueg.
On Physical Science in Schools
Authors: Barolay A.J.G.
Published: January 2009

Now that more science has become the popular educational cry, there is a danger of raising too great expectations of what physical science can do, and so of paving the way for a reaction against it when it is found not to yield the results unduly expected of it. This arises mainly from basing the claims of physical science to a school place upon an exaggerated estimate of the value of the knowledge imparted, and from not admitting it as an educative agent capable of filling a unique place in the educational course. On account of the popular belief in the ultimate practical, or bread-and-butter value of science teaching, it has been introduced in many cases as an attraction in a school prospectus without adequate means being provided for efficiently carrying it on. In such cases it often takes the form of the popular lecture illustrated by experiments which requires no great mental application on the pupil's part, gives him amusement, and relieves him for the time from some of his dry daily routine, but which fills him at the same time with a false and mischievous notion of what science is. It is of great importance that the true place and aim of physical science in schools should be clearly recognised, not merely by scientific men and educationists, but also by the intelligent general public, for it is only when such recognition is general that the means of equipping and maintaining science work will be forthcoming.
The Theory of Contours, and its Applications in Physical Science
Authors: Peddie W.
Published: January 2009

1. The word contour is largely used in ordinary language, but its meaning, when so used, is in general very different from its meaning as a scientific term. We speak of the contour of a hill, a cloud, a country, and so on; meaning usually a profile or an outline,—sometimes a particular outline only. Yet, even in this popular use of the word, we have an indication of its more exact significance. Thus, we see that the visible horizon, if we consider it to be a contour line, is the curve in which the earth's surface is met by its tangent-cone the vertex of which is at the observer's eye. The tangent-surface has a constant characteristic; and it is this possession of a distinctive property by all surfaces which give rise to contour lines, which furnishes the reason for the peculiar applicability of the method of contours to physical problems.
A Proof of Lagrange's Theorem
Authors: Miller T. Hugh
Published: January 2009

If y=z + xy(y) where x and z are independent, Lagrange's series gives the expansion of any function of y in terms of x. The coefficients of the powers of x may be found thus. Let ø(y) be the function to be expanded. Then by Taylor's theorem
Bifilar Suspension treated by the method of Contour Lines
Authors: Rainy Harry
Published: January 2009

The object of the following paper is to illustrate how readily some problems, which frequently occur in physical work, may be solved by the application of certain methods which are not very generally employed in mathematical investigations. The method on which the following proof is based is that of contour lines. These are a device enabling us to represent the third dimension on a plane, and are such that all the points on one contour line are at one and the same height above the level of the base plane. Probably the best known application of the principle of contours is to be found in the Survey Maps, where successive (closed) curves pass through all points at heights of 100, 200, 300, &c., feet above the sea-level; but they are of equal value in many physical diagrams, where they appear as equipotential lines, isothermals, lines of equal illumination, lines of equal force in a magnetic field, as well as in many other forms. In this paper they are employed as representing the third dimension merely. When contour lines are placed so as to indicate an equal rise in the intervals between each successive pair [or, in fact, according to any definite system], we can, with their aid, tell two things about a surface; the one—what is the steepness, or gradient, in any direction from a given point; the other—which is the direction of steepest slope at any point. The absolute steepness is measured by the amount of vertical motion per unit of horizontal motion of a point; i.e., by the tangent of the angle between the horizontal plane and the line in which the point moves. This is a fact with which everyone is acquainted ; every person knows what is meant by saying that the gradient along a line of railway is 1 in 50 or 1 in 500, as the case may be. To find the steepness at any point we merely need to know, in terms of the length of the line taken to represent unit steepness, the length of the line passing through the point in the given direction, and terminated by the two contours on either side of that point. The length of the line will then be inversely as the steepness.
Abstract of one of Euler's papers
Authors: Mackay J. S.
Published: January 2009

The paper is entitled Solutio facilis problematum quorumdam geometricorum difficillimorum, and is printed in Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom. xi., pp. 103–123. The volume is for the year 1765; the title-page is dated 1767.
Projective Geometry of the Sphere
Authors: Allardice R. E.
Published: January 2009

If A and B be two fixed points on a great circular arc and P a variable point on the arc, there are two and only two possible positions of the point P corresponding to a given valae of the ratio sinAP/sinBP, provided arcs measured in one direction from A or B be considered positive, and in the opposite direction negative; and these two points are antipodal.
On the Theory of Contours, and its Applications in Physical Science
Authors: Peddie W.
Published: January 2009

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines.

If any curve be cut by planes parallel to that of (x, y) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z-axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z-co-ordinate.
Notes on Integration by parts and by successive reduction
Authors: Gibson George A.
Published: January 2009

My object in the following notes is to call attention to some points in integration by successive reduction which may be of use in directing the choice of the particular form for the reduced integral in any given case. The remarks apply chiefly to the binomial differential x(α + bxn)pdx which has been discussed from a different point of view by Dr Muir in vol. iii., p. 100 of the Proceedings.
Question proposée an Concours Général pour la classe de Mathématiques Spéciales, Juin, 1886.
Authors: Aubert M. Paul
Published: January 2009

Étant donnés une surface du second ordre S et deux points A et B, on mène par le point B une sécante qui rencontre la surface aux points C, C′, et le plan polaire du point A au point D. Soient M et M′ les points of la droite AD rencontre les plans qui touchent la surface aux points C et C′. La sécante BD tournant autour du point B, on demands

1°. Le lieu décrit par les points M et M′.

2°. Ce lieu se compose de deux surfaces du second ordre, dont l' une est inépendante de la position du point B, et l' autre Σ dépend de la position de ce point. Chercher ce que devient la surface Σ quand, dans la construction qui donne les points de cette surface, on fait jouer au point A le rôle du point B, et inversement.

3°. Le point A restant fixe, déterminer les positions occupées par le point B quand la surface Σ n' α pas un centre unique à distance finie.
Note on a Formula in Quaternions
Authors: Allardice R. E.
Published: January 2009

The formula referred is the condition for the coplanarity of the extremities of four coinitial vectors; namely, if α, β, γ, δ, are the vectors, then

(See Kelland and Tait's Quaternions, p. 62.)

In the first place, it may be pointed out that these equations do involve one condition on the four vectors α, β, γ, δ. For, since the vanishing of a vector involves three conditions, and only the three ratios of the four quantities a, b, c, d, are involved, it is always possible to determine these four quantities so that the condition aα + bβ + cγ +dδ = 0 shall be satisfied. But, in general, the equation a + b + c + d = 0 will not be satisfied; and thus one condition is involved.
On a Proposition in Statics
Authors: Niven C.
Published: January 2009

It is known that a force acting along any line in space may be resolved into six components. In the most commonly employed resolution these are forces along three lines at right angles, and couples round these lines. But the six components may be taken to be forces along the six edges of a tetrahedron. It is the object of what follows to determine these forces.
On the most Economical Speed to drive a Steamer in relation to the cargo carried and coals consumed
Authors: Millar W. J.
Published: January 2009

The object of this communication is simply to show how a formula may be obtained which will indicate the most economical speed for a steamer in relation to cargo carried and coals consumed on the voyage.
On Vortex Motion in a rotating fluid
Authors: Chree C.
Published: January 2009

The object of the following paper is to consider the motion of one or more vortices in a compressible fluid, which is rotating as a whole with uniform angular velocity ω about an axis, taken as axis of z. To save space I shall when possible refer for results to a previous paper in the Proceedings, distinguishing the equations of that paper, Vol. V.; pp. 52–59, by the suffix a.
On the expression of a symmetric function in terms of the elementary symmetric functions
Authors: Allardice R. E.
Published: January 2009

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.
Solutions of two geometrical problems
Authors: Mackay J. S.
Published: January 2009

The two problems are :—

1. To divide a given straight line internally and externally so that the ratio between its segments may be equal to a given ratio.

2. To divide a given straight line internally and externally so that the rectangle under its segments may be equal to a given rectangle.
On the condition that the straight line lx+my+nz=0 should be a normal to the conic (a, b, c, f, g, h)(x, y, z)2=0 the co-ordinates being trilinear
Authors: Pinkerton R. H.
Published: January 2009

The condition in question may be found by using the following theorem:-

If the equation in trilinear co-ordinates
Some Geometrical Theorems
Authors: Allardice R. E.
Published: January 2009

The greatest line joining two points in the perimeter of a polygon is a side or a diagonal of the polygon.

For (fig. 2) PS is obviously less than one or other of the lines PQ, PR. Hence if AB, CD (fig. 3) are any two finite lines, L and M any points, one in each of these lines, LM is less than LC, or LD and LC is less than CA or CB. From this the theorem follows at once. The theorem and the above proof apply to crossed and gauche polygons.
The solution of the equation √{(x−a)(x−b)}(x−c)(x−d)}=e
Authors: Dickson J. D. Hamilton
Published: January 2009

If P=a+b−c−d, and K=cd−ab, the solution of the equation

may be most readily found by putting (x−a)(x−b)=z2; whence (x−c)(x−d)=z2+Px+K, and therefore =z2−2ez+e2, and finally x is given by the equation
Some relations between the orthic and the median triangles
Authors: Pressland A. J.
Published: January 2009

Let ABC be the triangle, X, Y, Z the feet of the altitudes, H the orthocentre, A′,B′,C′ the mid points of the sides.

Let ZY meet A′B′ in D, A′C′ in D′, B′C′ in R:

ZX meet B′C′ in E, A′B′ in E′, A′C′ in S:

XY meet C′A′ in F, B′C′ in F′, A′B′ in T.
Note on an approximate fractional expression for the expansion of (1+x)n to any odd number of terms
Authors: Steggall J.E.A.
Published: January 2009

In Bonnycastle's Arithmetic a rule is given for finding any root of a number by approximation which in substance reduces to this statement:—If a and b are nearly equal, then the nth root of a/b is nearly equal to
On the history and degree of certain geometrical approximations
Authors: Pressland A.J.
Published: January 2009

According to Luther's translation 1 Kings vi. 31 should read— “At the entrance of the choir he made two doors with pentagonal door-posts.” This is probably a wrong translation, for nowhere on Asiatic monuments of this time has a pentagon been found. Prof. A. Merx in Cantor's Vorlesungen über Geschichte der Mathemalik, Vol. i., p. 91.

It is probable that the construction of the regular pentagon is due to the Pythagorean school as a consequence of I. 47. (Ibid, p. 151.)
On the history and degree of certain geometrical approximations
Authors: Pressland A.J.
Published: January 2009

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.

On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.
On the Contact-Property of the Eleven-point Conic
Authors: Allardice R.E.
Published: January 2009

It is sometimes the case that geometrical theorems, which are usually enunciated as properties of the triangle or the quadrilateral, may be stated more succinctly, and in a form that better suggests generalisation, as properties of the complete quadrilateral. Thus the theorem that is usually stated in the following form, “The three straight lines that join the middle points of opposite sides and the middle points of diagonals of a quadrilateral are concurrent and bisect one another at the point of concurrence,” may be enunciated more simply as a property of the straight lines that join the middle points of the three pairs of opposite sides of a complete four-point; and the latter form of enunciation suggests how the theorem may be generalised by projection. So too the figure that consists of four Points, any one of which is the orthocentre of the triangle formed by the other three, is better described as a complete four-point in which opposite sides intersect at right angles, (which may be called an orthic four-point).
Experimental Introduction to the Study of Magnetism
Authors: Knott C.G.
Published: January 2009

The aim of the scientific teacher is to teach the pupil how to think along scientific lines. By a suitable presentation of the facts of experience he should lead the mind of the learner to form almost intuitively the scientific law or generalisation which embraces them all. We may of course start with the law or formula, and develope it mathematically into all its ramifications. But that reduces itself to mere analytical skill. If carried out faithfully in the elementary teaching of science, it would tend to give the learner an erroneous conception of the whole method of scientific investigation and the meaning of scientific law.
On a surface of the third order
Authors: Allardice R.E.
Published: January 2009

In the second part of Professor Chrystal's Algebra (Exer. V., No. 4) the following exercise is given:-

If 2xyz-x2-y2-z2+1=0, and x,y,z are all real, then all, or none, of the quantities x,y,z lie between-1 and + 1.
On the Solution of Non-linear Partial Differential Equations of the Second Order
Authors: M'Cowan John
Published: January 2009

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.
Integrals of the form and allied Integrals
Authors: Gibson George A.
Published: January 2009

1. If θ be the eccentric angle of a point P on an ellipse whose semi-axes are a and b and if x, y be the rectangular co-ordinates of a point Q, then PQ2 = (x − acosθ)2 + (y − bsinθ)2. When a = b and x2 + y2 = f2, PQ2 = f2 2afcosθ + a2, if the line from which θ is measured passes through Q. The integral

is a well-known one. I propose in the present paper to consider the more general form which the integral takes when P lies on an ellipse.
Matthew Stewart's Theorem
Authors: Mackay J.S
Published: January 2009

In 1746, when he was a candidate for the Chair of Mathematics in the University of Edinburgh rendered vacant by the death of Maclaurin, Matthew Stewart published his first work, Some General Theorems of considerable use in the higher parts of Mathematics. In the preface to it he states that “the theorems contained in the following sheets . … are entirely new, save one or two at most,” but he does not specify the two. They are
A Problem of Lewis Carroll's, and the rational solutions of a Diophantine Cubic
Authors: Tweedie C.
Published: January 2009

§ 1. In the Life and Letters of Lewis Carroll occurs the following extract from his Diary:—

“Dec. 19 (Sun).—Sat up last night till 4 a.m., over a tempting problem, sent me from New York, ‘to find 3 equal rational-sided rt.-angled ∆&s.’ I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three.' (v. page 343.)
Solution of the Cubic Equation
Authors: Collignon M. Edouard
Published: January 2009

§ 1. The roots of the cubic equation

are given by the formula

This solution is of little practical use when the roots of the cubic are all real and unequal, that is, when is negative (the-Irreducible Case of Cardan's Solution).
The determination of the centre of gravity of a circular arc by reference to a principle of Dynamics
Authors: Crawford G. E.
Published: January 2009

Mr H. Poincaré, in a recent work (la Valeur de la Science) speaks of the interest attaching to proofs of theorems in pure geometry by arguments derived from mechanical or physical considerations. The following is a case in point.

To find the centre of gravity of a circular arc by reference to a principle of dynamics.
Proofs of an Inequality
Authors: Muirhead R. F.
Published: January 2009

§ 1. The inequality of the Arithmetic and Geometric Means of n positive quantities has been proved by many different methods; of which a classified summary has been given in the Mathematical Gazette (Vol. II., p. 283). The present article may be looked on as supplementary to that summary. It deals with proofs that belong to a general type, of which the proof given in the Tutorial Algebra, §205, and that given by Mr G. E. Crawford in our Proceedings, Vol. XVIII., p. 2, are very special limiting cases. Proofs of the type in question consist of a finite number of steps, by which, starting from the n given quantities, and changing two at a time according to some law, we reach a new set of quantities whose arithmetic mean is not greater, and whose geometric mean is not less than the corresponding means of the given quantities.
Conditions for the Reality of the Roots of an n-ic
Authors: Peddie W.
Published: January 2009

The homogeneous real linear transformation in n variables is such that, when these variables are used as a set of mutually rectangular coordinates, an n-dimensional sphere is transformed into an n-dimensional ellipsoid; n mutually rectangular radii of the sphere become the n, mutually rectangular, principal radii of the ellipsoid. When these principal radii have not been rotated from their original directions, the transformation is said to be pure, or irrotational. Since these radii are necessarily real, the roots of the n-ic for the determination of the n principal elongations are necessarily real.
Notes on the Apollonian Problem and the allied theory
Authors: Dougall John
Published: January 2009

1. This paper contains a number of investigations, more or less connected, on the theory of systems of circles. In such a well-worn field one does not expect to have hit upon much that is absolutely new, but it may be hoped that there is sufficient freshness of treatment to give the paper some interest even where it deals with results already known.
A Symbolic Method in Geometrical Optics
Authors: Ross Edward B.
Published: January 2009

The formulae given in Herman's Optics, pages 80, 82, 98, 111, and called Cotes's formulæ, are a little difficult to grasp, and do not lend themselves to manipulation. The notation explained below is useful as a mnemonic. I think it also renders the proofs simpler.
On the Envelope of the Directrices of a System of Similar Conics through three Points
Authors: Allardice R. E.
Published: January 2009

The problem was proposed by Steiner of finding certain loci and envelopes connected with a system of similar conics through three points. In previous papers I have found the locus of centres and the envelope of asymptotes of such a system of conics, also the envelope of axes, and the locus of foci. I now propose to discuss the envelope of the directrices.
On a Class of Surfaces
Authors: Miller John
Published: January 2009

The surfaces here considered were first discussed by Monge as surfaces whose normals are tangents to given developables. Under the name general surfaces moulures, Darboux treats them as the surfaces traced out by a fixed curve on a plane which rolls on a developable. When the developable is a cylinder he gives the general coordinates in his Leçons sur la Theorie Generale des Surfaces (Volume I., page 105). This led me to take up the more general case, but I later found that Darboux had also considered this in an ingenious and elegant manner in his Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes (Tome 1, pages 26–34). Perhaps this quite difFerent discussion will present some interesting points in analysis.
End-to-end Focal Chords in an Ellipse
Authors: Taylor David G.
Published: January 2009

The relation between the eccentric and focal angles of P (Fig. 1) is

or, more compactly,

where
The system of cubic curves circumscribing two triangles and apolar to them
Authors: Milne William P.
Published: January 2009

It is very seldom that any properties of a pencil of cubics can be discovered beyond some very general ones. The above case yields, however, some interesting results. It presented itself to me when I was considering the apolar generation of cubic curves which was published in the Proceedings of the London Mathematical Society (Ser. 2, Vol. 9, Part 3), and which I shall have frequent occasion to refer to in the present paper.
An Analytical Study of Plane Rolling Mechanisms
Authors: Stokes G. D. C.
Published: January 2009

In the Appendix III. of his “Mechanics of Machinery,” Le Conte attacks the problem of rolling curves by an elegant analytical method in which the cartesian forms of the involute and the trochoids are derived from elementary differential equations. Weisbach(“Mechanics of Engineering and of Machinery,” Vol. III., chap. II.), while using mainly the geometrical method, discusses one application analytically (see §3 of this paper) in which polar forms are introduced. Many other writers, for example Barr (“Kinematics of Machinery,” chaps. III., IV.), deal wholly with the geometry of the subject.
Further Investigations on Circular Cubics and Bi-Circular Quartics
Authors: Milne William P.
Published: January 2009

Last session (1910–11) I contributed two papers investigating the Focal properties of Circular Cubics and Bi-Circular Quartics by the methods of pure geometry. I find on further investigation that not only can the properties of the individual Focal Circles be directly found by the method there given, but the mutual properties of the Focal Circles can be obtained by an extension of the same method. Several geometrical results, which as far as I know are new, are obtained incidentally in the discussion.
Note on the Researches of Maclaurin on Circular Cubics
Authors: Teixeira F. Gomes
Published: January 2009

In the works known to us which contain historical or bibliographical information about circular cubics no sufficient indication of the researches of Maclaurin is found. Yet many of the classic propositions connected with these curves are due to this eminent geometer, who investigated constructions for unicursal circular cubics, for certain non-unicursal circular cubics, and for the special circular cubics now known as the trisectrix of Maclaurin, the oblique cissoid, and the strophoid. His name does not appear even in the list of writings on the strophoid published by Tortolini and Günther.
On the Summation of lr + 2r + 3r + … +nr
Authors: Donaldson J. A.
Published: January 2009

We shall use the notation Σnr= lr + 2r+…+ nr. Mr A. J. Gray suggested to me that n(n+ 1) is always a factor of Σnr, and that, in addition, 2n + 1 is a factor when r is even.
A method of finding (i) the double points of a unicursal curve, (ii) unicursal quartics with three given double points
Authors: Bell R. J. T.
Published: January 2009

The equations

where f1(t), f2(t)φ(t) are polynomials in t, determine a unicursal curve. If α and β can be found so that

then the point (α, β) is on the curve and is given by two values of t, t1 and t2, which are the roots of the equation at2 + 2bt + c =0.
Integral Equations and the Determination of Green's Functions in the Theory of Potential
Authors: Carslaw H. S.
Published: January 2009

In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—

(i) In the interior of S, it satisfies ∇2V = 0.

(ii) At the boundary of S, it vanishes.

(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).
Geometrical Constructions for Refraction and Reflection in a Prism
Authors: Robertson David
Published: January 2009

General Construction for Refracted Ray. The two-circle method of finding the direction of a ray refracted at a plane surface is very old, but seems to be now almost forgotten. It is particularly convenient when the refraction of several rays is to be determined, and its application to the case of a prism is especially elegant and leads to a simple self-contained proof of the condition for minimum deviation.
Note on Whittaker's Solution of Laplace's Equation
Authors: Copson E. T.
Published: January 2009

§1. Whittaker has shewn that a general solution of Laplace's Equation

may be put in the form

where f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.
Peaucellier's Cell and other linkages in non-euclidean Geometry
Authors: Sommerville D. M. Y.
Published: January 2009

There are certain well-known linkages for effecting the transformation of inversion, and, incidentally, by inverting a suitably situated circle, for producing straight-line motion. Reference may be made to the classic lecture by A. B. Kempe, “How to draw a straight line” (London: Macmillan, 1877). It is the object of this paper to call attention to the fact that these linkages have the corresponding property in non-euclidean geometry.
Some Criticisms of Robert Simson by Sir T. L. Heath
Authors: Gibson George A.
Published: January 2009

Sir T. L. Heath's translation of The Thirteen Books of Euclid's Elements, with its Introduction and Commentary, is not merely a worthy tribute to the lasting merits of Euclid's work, but is at the same time a most valuable history of elementary geometry; the language in which he describes the character of Camerer's edition of Euclid's first six books is even more applicable to his own: “No words of praise would be too warm for this veritable encyclopaedia of information.” There may, however, be room for difference of opinion on matters of detail, and I propose in this note to call attention to one or two passages in which I think he is in error in his criticism of Simson, whose edition of Euclid formed the basis of so many English text-books and kept alive the traditions of Greek geometry in this country long after Euclid's Elements had disappeared as a text-book on the Continent.
A Theorem on the Integral of Stieltjes
Authors: Hyslop J.
Published: January 2009

In a recent paper Mr J. M. Whittaker has given the following Theorem:—

If ψ (x) be the indefinite Riemann integral of a bounded positive function g(x), and if f(x) be any bounded function, then the equation

is true whenever either side exists.
Local Electromagnetic Fields
Authors: Bateman H.
Published: January 2009

§ 1. The radiation of electromagnetic charges from a moving point.

Let the electric charge associated with a moving point Q at time s be f(s). This charge is supposed to vary on account of the radiation of electric charges from Q in a variable direction, which, at time s, has direction cosines l (s), m (s), n (s) Using ξ(s), η(s), ζ(s) to denote the coordinates of Q at time s, and defining the effective time τ by means of the equation

the electromagnetic field may be specified by means of the potentials

where
On the stationary eccentricity of a system of four-tangent conics
Authors: Edwards F. E.
Published: January 2009

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is then

where k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.
On the Integration of the Canonic Equations and the Principle of Duality
Authors: Stokes G. D. C.
Published: January 2009

The object of this paper is to essay an analytical statement of the reduction of the integration of a canonic system of differential equations (into which time does not enter explicitly) to that of the partial differential equation of Jacobi and Hamilton; and to illustrate the principle of duality by an outline of the solution for the problem of two bodies both by the standard form of the equation referred to and by the analogous form which that principle involves. Most statements of the reduction are verifications and somewhat obscure the symmetry of the canonic form. The shortest procedure, of course, is by means of the well known theorem of Jacobi, and this verificatory method is followed by Tisserand, Charlier and Appell. Poincaré gives a proof depending on a simple form given by him to the conditions for a canonical change of variables, but again the statement lacks analytical form. The essentials of this proof will be given here, but in an entirely different way. An analytical treatment of the subject has been given by Professor L. Becker in his class lectures at Glasgow, but it has not been published.
Classification of Geometries with Projective Metric
Authors: Sommerville D. M. Y.
Published: January 2009

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.
The Locus of the Straight Lines which intersect Three Given Lines
Authors: Bell Robert J. T.
Published: January 2009

The following note indicates how the equation to the locus of the straight lines which intersect three given lines may be obtained by the use of the conditions that three planes should have a line of intersection.

Three given planes

have a line of intersection if any two of the determinants in the scheme

are equal to zero.
The Factors of (a, b, c, f, g, h)(x, y, z)2–λ(x2+y2+z2)
Authors: Bell Robert J. T.
Published: January 2009

f λS is the product of two factors of the form αx + βy + γz if λ is a root of a discriminating cubic

A well-known proof of the reality of the roots of the cubic is as follows :—

Write