The following is a generalization of a theorem stated by Professor H. S. Uhler and demonstrated by the writer in the American Mathematical Monthly of October 1927.

All the commonly used rules for the approximate quadrature of areas, such as those of Cotes, Simpson, Tchebychef and Gauss, are based on the assumption that y can be expressed as a rational integral function of x with finite coefficients. A tacit assumption is thus made that is not infinite within the range considered, and it is therefore hardly a matter for surprise that the degree of accuracy obtainable by the use of these rules in the case of a curve which touches the end ordinates is very poor.

In the short sketch which I propose to give of the History of Mathematics in Scotland up to the end of the 18th century I must limit myself mainly to the work of the Universities. An adequate treatment of the subject would involve considerations of a general educational character that would range over the relations of the school to the University, the distribution of the various subjects of study and the place of mathematics in the educational system; but it is, of course, impossible to undertake such an extensive investigation at present, though it seems to me that an investigation, with special reference to mathematics, is greatly needed and might form the subject of a research that would be of real value as a contribution to the development of educational ideas. It would be improper, however, to omit all reference to school mathematics, since the school conditions determine, to a considerable extent, those of the University, as current discussions in Scotland clearly show, even though a sound appreciation of the relations between school and University may at times be lacking.

The centre of interest now shifts from St Andrews and Edinburgh to Glasgow. The troubles that afflicted Scotland during the 17th Century bore heavily on Glasgow University and more particularly on the position of Mathematics in the University; but in 1691 a distinct Professorship of Mathematics was founded, and from that date the old system of Regents disappeared from Glasgow so far as Mathematics was concerned. The first occupant of the Chair was George Sinclair, who is now chiefly remembered by the controversy in which James Gregory held up Sinclair's Treatise Ars nova et magna to ridicule. It is not fair however to take Gregory's pamphlet as a final estimate of Sinclair's contributions to science; Sinclair laid himself open to attack, but he rendered great service to the mining industry of Scotland and deserves the gratitude of posterity in spite of his many eccentricities. His contributions to mathematics however are of no importance, but during his tenure of the Chair the number of students grew rapidly and the new professorship made a good start.

Let bmn be a positive function of m and n which decreases steadily with n, so that bmn ≥ bm, n+1 for all values of m and n. Assume also that |am1 + am+2 + … + bmn| < K for all values of m and n, K being finite. Denote bySmn the sum of the first n terms in the first m rows of the double series

We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:

Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.

Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric is

Two proofs of this result are given, in §§ 4 and 5.

This paper shows briefly how, for a singly infinite family of curves on a given surface, the fundamental properties of the family at any point are associated with three central conies determined by the curves, in a manner resembling that in which the curvature properties of a surface at any point are associated with Dupin's indicatrix. The differential invariants employed are the two-parametric invariants for the given surface.

Let umn, vmn be functions of m and n, vmn being real and positive for all positive values of m and n. Suppose that either vmn increases steadily to infinity with n, or that umn both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by wmn, and assume that wmn exists for every value of m, being denoted by lm. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then lm → l as m → ∞.

In comparison with the general plane quartic on the one hand, and the curves having either two or three nodes on the other, the uninodal curve has been neglected. Many of its properties may of course be deduced from those of the general quartic in the limiting case when an oval shrinks to a point or when two branches approach and ultimately unite. The modifications of properties of the bitangents are shewn more clearly by Geiser's method, in which these lines are obtained by projecting the lines of a cubic surface from a point on the surface. As the point moves up to and crosses a line on the surface, the quartic acquires a node and certain pairs of bitangents obviously coincide, viz. those obtained by projecting two lines coplanar with that on which the point lies. A nodal quartic curve and its double tangents may also be obtained by projecting a cubic surface which has a conical point from an arbitrary point on the surface. Each of these three methods leads us to the conclusion that, when a quartic acquires a node, twelve of the double tangents coincide two and two and become six tangents from the node, and the other sixteen remain as genuine bitangents: the twelve which coincide are six pairs of a Steiner complex.

I. If the Jacobian sextic of a pencil of binary quartics is known, the pencil itself is determined in five ways. The explicit determination of the pencil in terms of the irrational invariants of has been effected by Stephanos.

There are two known cases in which a pencil of quartics admits of rational determination from its Jacobian. In these, the rationally determinable pencil differentiates itself algebraically from its remaining four co-Jacobian pencils.

If L, M, N denote Prof. Study's Dual Coordinates of a straight line (see Proc. Edinburgh Math. Soc., 44 (1926), 90–97), any (homogeneous) equation F (L, M, N) = 0 must define a certain system of lines. By the nature of dual numbers we must have

where U and V are functions of l, m, n, λ, μ, ν, the ordinary (Pluckerian) coordinates. Since F = 0 implies U = 0 and V = 0 the system of lines is a congruence. But it is a congruence of a very special kind, whose nature will now be considered.

The operations of differentiation, ordinary and central differencing, divided differencing, and forming linear combinations of the results of these, have in common the distributive, associative and commutative properties, and the further property that when performed on a general polynomial of the nth degree they produce a polynomial of the n−1th degree, while when performed on a constant they yield zero.

The matrices considered in the following note are non-singular, and are related to a given matrix A having elements taken from the field of positive and negative integers and zero. The invariant properties of such a matrix A, under multiplications by matrices of determinant equal to unity, can be formulated, as is well known, in terms of the “elementary divisors” of the determinant |A|. Thus if A is of the nth order, and p is a prime occurring in |A| to the power hn, in the H.C.F. of the first minors of |A| to the power hn−1, in the H.C.F. of the second minors to the power hn−2, and so on, h0 by convention being zero, then the first differences of the h's,

are invariant under the transformations considered, and it is known that r ≥ er−1. The numbers

where the product includes all prime factors of |A|, are called the elementary divisors of |A| and

The theorem is well known. So also is the theorem that if concerning a determinant Λ and its reciprocal expressed by means of cofactors Aij of aij. Not quite so well known is the Cayley Hamilton theorem that a matrix X =[xij] satisfies its own characteristic equation

Unlike as these three results are, they nevertheless can be looked upon as particular phases of a general theorem concerning a matrix differential operator acting upon a function of a matrix X or its transposed.

The derivation of the 4-nodal Cubic Surface and the Quartic Surface, of which it is a particular case, are well known: certain new results of interest from the point of view of symmetry, and extension to n-fold space, are provided by the symbolic algebra. In particular a simple proof is given, in § 2, of the theorem that a symmetry exists between the four vertices of the tetrahedron and the fifth point whose locus is the cubic surface, and this property can be extended to the case of n + 1 points in n-fold space with one additional point.

The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that

1. The cardinal function is the interpolation function

which takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that

(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.

§1. The Cardinal Function of Interpolation Theory is the function

which takes the values an at the points x = n. Ferrar has recently proved

Theorem1. If are convergent, C(x)is an m-function3for

This means that C(x) is a solution of the intergral equation

Ferrar's proof deals with functions of a real variable and involves some rather difficult double limit considerations.