1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.

There is a group of Tauberian theorems of which the simplest isone due to K. Ananda Rau [Theorem 2 of the paper numbered 1 inthe list of references at the end of the note]. More complicatedtheorems of the same group are discussed in a paper by S.Minakshisundaram and myself to be published by the LondonMathematical Society [4].

The purpose of this brief note is to draw attention to a type of inverse factorial series which, so far as the writer can judge, has not been intensively studied. The central difference formulae of interpolation and the corresponding infinite series in central factorial polynomials have in the past received much attention, and so also has the ordinary inverse factorial series

which from the time of Stirling (Methodus Differentialis, 1730) has been known to be of value in transforming slowly convergent series into series of more rapid convergence.

§ 1. Corresponding to any partition

which we denote by [a1, . …, ak] or briefly by [a], of the integer n, we can construct a shape which has a1 spaces in the first row, a2 in the second row, . …, ak in the kth and last row. Thus the shape corresponding to the partition [5, 3, 3, 2] of 13 has the form:

The theory of four particular linear forms, or matrices of k columns and 2k rows, occurred to me many years ago in an attempt to study the invariants of any number of compound linear forms, or subspaces within a space of n dimensions. In what follows, the invariant theory is given, and its significance for a study of the general matrix of k rows and columns is suggested. The collineation used in §4 was considered by Mr J. H. Grace, who emphasized the importance of the k cross ratios upon transversal lines of four [k−1]'s in [2k−1]. It seemed appropriate to examine these cross ratios which are irrational invariants μi, of the figure of four such spaces, and to work out their relation to the known rational invariants Xi. The main result is given in § 5 (7). In § 5 (10) it is shewn that the harmonic section of a line transversal of the four spaces exists when a linear relation holds between the invariants.

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.