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.

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.

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.

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.

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.

In this note I prove the following result:—

Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.

In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

We have shown in [1] that under certain conditions the definite integral may be approximated by a determinantal ratio. It is our object now to develop the theory when C (x) is a polynomial, showing the relation to the continued fraction form for . In particular we shall give various forms for the approximants, and an integral form for the numerator.

1. This note gives an asymptotic evaluation of an integral of the form

as n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.