In theory of polarizing operators in invariants the operator

where X = [xij] is an n × n matrix of n2 independent elements xij, holds an important place. Acting upon particular scalar functions of X, namely the spur or trace of powers of X, or of polynomials or rational functions of X with scalar coefficients, it exhibits (Turnbull,. 1927, 1929, 1931) an exact analogy with results in the ordinary differentiation of the corresponding functions of one scalar variable. Turnbull denotes this operation of trace-differentiation under Ω by Ω8; and we shall follow him. Our purpose is to show how, with a suitably modified Ω, the results may be extended to the case of symmetric matrices X = X′ having ½n(n + 1) independent elements.

1. It has been shown (1; 2, 136) that if Sr, ar, hr denote respectively the symmetric functions , Σλ1 λ2…λr, and the homogeneous product sum of degree r of the latent roots λ1, λ2, …, λn of the matrix X = [xij] then

In a recent paper1 γ-matrices were constructed summing the binomial series at isolated points to its right value. It is assumed that the reader is able to refer to this paper and is familiar with its notation and terminology.

The object of this note is to give the construction of γ-matrices which sum the binomial series at an isolated point to an arbitrary given value, while inside the circle of convergence the generalised sum is necessarily the right value.

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:

Let the expression be given; introduce, for c, a linear form in and obtain

If the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equations

where Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the set

where Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

Vajda's paper in this volume has suggested to the author the following problem:

Let A be a n × m matrix, m≦ n, let B be a m × n matrix, and let M = I – AB, where I is the unit matrix of order n. Given A, to find B such that of the n latent roots of M′M, k are unity, and the remaining n — k are zero.

The geometry associated with the five invariants of two quadrics is well known. There appears to be an omission, however, in the treatment of the Φ-invariant. Salmon gives the vanishing of Φ merely as a necessary condition for the possibility of the construction of a tetrahedron self-conjugate for one of the quadrics and having its six edges tangential to the other. Sommerville proves its sufficiency, and shows that this condition is poristic, giving rise to two systems of ∝ tetrahedra of the requisite type. No investigation appears to have been made of the locus of the vertices of the ∝ tetrahedra of each system and the dual problem regarding the nature of the developable surface arising from their ∝ faces.

Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise, p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p + 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. K will denote both the set of all skew-symmetric matrices and the ½p(p – 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix. Dα will denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.

A useful test for uniform convergence is that first established by Buchanan and Hildebrandt [4] which is as follows.

(A) “If a sequence fn (x) of monotonic functions converges to a continuous function f(x) in [a, b] then this convergence is uniform.”

In §1 of this paper it is shown that this test is included in a sequence of theorems, each of which establishes a type of uniform convergence. The first is a well-known topological theorem on limit sets, the second is a result on the limits of rectifiable arcs, the third is a generalisation of (A) due to Behrend [3], the fourth is (A) itself, the fifth is a one-sided version of Bendixson's test and the sixth is Bendixson's test.