In a recent paper some general properties of γ-matrices were proved and Dienes' theorem on regular γ-matrices extended to semiregular γ-matrices and the binomial series. In section 2 of this paper the previous results will be extended to certain classes of Taylor series. Section 3 gives some new results on Borel's exponential summation, and section 4 introduces matrices efficient for Taylor series on the circle of convergence and others efficient for Dirichlet series on the line of convergence. A knowledge of the definitions and results of the paper mentioned above is assumed.

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

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.

This paper deals with the multivariate normal distribution in the symmetrical case when the frequency function may be reduced to

This is the usual distribution occurring for a series of similar independent observations or for the firing of shots at a target.

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.

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the form

where x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.

I propose to prove the following theorem.

With n > 2, the (n + 2) equations derived from the matrix

where

by equating to zero all (n + 1)-rowed determinants from the matrix ‖Δ‖ are equivalent to only two, one of which is linear in li (i = 1, 2, …, n) and the other is homogeneous and quadratic in a certain n – 1 of li (i = 1, 2, …, n); the elements of the matrix are real; (r, s = 1, 2, …, n) and d is arbitrary.

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series by

Then the allied series is

Let P be an n-rowed skew-symmetric matrix of rank 2r with elements out of an infinite field F. Denoting by x, y columns of n variables (indeterminates over F) xv, yv (v = 1, …., n), and by x′, y′ the corresponding row matrices, we consider the skew-symmetric bilinear form y′Px. It is well known that for every P a regular homogeneous substitution over F can be found so that

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.

I consider an n-dimensional generalized metric space Sn with coordinates xi(h, i, j, k… run from 1 to n throughout), with each point of which is associated a contravariant vector-density with components ui and weight p, called the element of support. The unit vector in the direction of the element of support has components denoted by li.

1. Introduction. In an earlier paper1 the function

was considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the form

where the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.

The present paper is concerned with the “logarithmetic”, or arithmetic of shapes of non-associative combinations as defined by Etherington in ref. (1). The shape of a non-associative product is defined as “the manner of association of its factors without regard to their identity”. Thus, for a binary non-communicative operation, the products ((AB)C)D and ((BA)C)D and ((AA)A)A all have the same shape, while D((AB)C) has a different shape. The sum of the two shapes a and b is defined as the shape of the product of two expressions, of shapes a and b respectively, in the original system of non-associative combination. The product of two shapes a and b is defined as the shape of any expression obtained by replacing every factor in an expression of shape b by an expression of shape a. It is readily shown that these definitions are unambiguous.

In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.

It has been proved by CAYLEY that if x11, x12, x21 … are independent variables, x = det (xik), ξ = det (ξik), (i, k = 1, … n) where ξik =∂/∂xik then by formal derivation ξxα = α(α + 1)…(α + n − 1)xα−1. This is a special case of the formula

where m=1,…,n and with i = i1,..im; k= k1,…km and xi,… is the algebracial complement of i = i1,..im; k = k1,…km, in .

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 result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

In this paper, I introduce a class of arrangements called arrays of strength d and discuss methods of constructing them with the help of finite geometrical configurations and algebraic groups involving elements of a Galois field. The definitions of arrays of strength d and other configurations that are used are given below.