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.

A form of integration of tensors will be introduced here, which will preserve the character of a tensor when so integrated.

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma has

and points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integral

which was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.

This paper has to deal with some topics suggested by a theorem which is notorious in the theory of quadratic forms, namely that the roots of the characteristic equation of any real quadratic form are all real. For a proof of this theorem, the reader may consult Fcrrar (1), p. 146.

Many-valued or non-Aristotelian calculi of propositions (logics) were originally introduced by generalisation of the truth-table method. It was known by the end of the nineteenth century that ordinary “binary” formulae of the calculus of propositions, such as

could be verified directly by means of the truth-table:

although the terminology and symbolism used were different.

We shall show in this part the relation of generalised C.F.'s to ordinary C.F.'s, in the main confining our attention to Stieltjes type fractions. Moreover we shall bring out the part played by Parseval's theorem in our development of the subject, and a property of extremal solutions of the Stieltjes moment problem given by M. Riesz.

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

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive (i.e. non-zero) coefficients of P, the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p. We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S (a) for the elementary divisors associated with p to be simple, and (b) that there is only one elementary divisor associated with p.

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.

We develop here the recurrence relations for the generalised C.F.'s introduced in Part 3 (Shenton 1956). In the main the discussion will be limited to second order C.F.'s, but results for higher orders will be given when these are not complicated.

1. Let

be an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notations

describing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).

1. Prof. E. T. Copson has discussed the well-known problem of a circular disc kept at a constant potential Vo in an external field of potential Φ by reducing it to the solution of two integral equations. Tho solution is however fairly simple if we use oblate spheroidal co-ordinates. This is due to the fact that in this system of coordinates the disc can be represented in terms of one co-ordinate only. This method is applied to the above problem and Copson's results are obtained. The solution when Vo is not constant, but any surface function of the disc, is also obtained.

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.

Titchmarsh and others [4] havo studied integral transform pairs of tho type

showing how such pairs may be generated as a consequence of the relation which holds between the Mellin transforms of the kernels.

Integral equations are obtained with nuclei (l — zt/a)2n and (z–t)2n which are satisfied by characteristic solutions of the transformed Lamé-Wangerin equation of order n, and each of the two characteristic solutions is expressed in terms of the other by a contour integral.

The following problem is a slight generalisation of one posed partly solved by H. Nagler.1 We shall use A* for the conjugate transpose of a matrix A. A projection is an idempotent matrix, its latent roots consist of units and zeros.

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

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.