The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third order

are permutation matrices. It is convenient to denote them by

where the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

In this paper we consider the correspondence between tangential quadrics of [3] and points of [9]. Godeaux has considered this geometrically, with the object of obtaining a representation for a twisted cubic of three dimensions. We have considered it from a standpoint more algebraic than that of Godeaux, with particular reference to the types of pencils of quadrics that correspond to special lines of [9], and to the interpretation in [9] of the fact that the condition for a net of quadrics to be part of the polar system of a cubic surface is poristic.

An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relations

between k generating operations S1, …., Sk of , such that every other relation between S1, …., Sk is an algebraic consequence of (1).

The number of groups for which abstract definitions are actually known is relatively small, but a remarkable feature of the results already obtained is the extreme simplicity of the relations (1) in the case of several groups of quite high order. This fact constitutes an additional incentive to the search for abstract definitions, and many elegant results have doubtless yet to be discovered.

It is well-known that if v≧–½ and

then ø (x) is said to be self-reciprocal in the Hankel transform and may be described as Rv. If v = ±½(1) reduces to the Fourier sine or cosine transform. Functions of these two classes may be described as Rg and Re.

In a list of pairs of reciprocal functions, G. A. Campbell gives the example that

is Rc, but there does not seem to be any explicit reference to this function in the literature. It suggests that there is a corresponding function which is Rv.

In an earlier “Note on Selection from a Normal Multivariate Population” the author considered the transformation of an n-variate normal population induced by “selection” operating to alter the parameters of frequency in a subset of p variates, while preserving normal frequency in the subset. The results were new in notation only; in substance they were first found by K. Pearson.

Following the papers by H. W. Turnbull and J. Williamson, I have verified that the 122 forms, of the system of two quaternary quadrics and are actually irreducible. The original 1917 system contained 125 forms, which Williamson reduced by three. The present verification shews that no further reduction is possible. The proof was carried out as follows. I first constructed the whole system in canonical form with and for the two quadrics, and then listed the degrees in the coefficients and variables u, p, x of these concomitants. I next made Diophantine equations between these degrees for testing the supposed reducibility and found them to be impossible, except for Williamson's reduced forms.

It is known that an induced matrix of an induced matrix is expressible as the direct sum of invariant matrices, or more generally that an invariant matrix of an invariant matrix can be expressed as a direct sum of invariant matrices. The spurs of the irreducible invariant matrices of a given matrix A = [ast], are the S-iunctions of the latent roots of A.

I begin by recalling the well known definitions for summability by the methods of Cesaro and Riesz.

The series Σan is said to be summable (C, K), k> – 1, to the sum s if, as n → ∞

where

and is defined formally by the relation

The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal curve of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in [n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem—that each of the sixty ways of arranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem—is retained in the following extension to [n].