In a recent paper I have discussed the families of quadrics in [2n] which are obtained by causing the members to have the greatest possible number of fixed [n – 1]'s or “generators.” It was found possible to fix four [n – 1]'s in general position; the family of quadrics through these possessed a “base” variety, common to all the members, which consisted of a highly degenerate Vn–1. Here I consider the same problem for quadrics in [2n + 1], find how many generators may be assigned arbitrarily and discuss the common part of the quadrics which pass through such generators.

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.

The physical observations that lead to quantitative physical theory are “pointer-readings.” The observational data consist of statements to the effect that, when one given set of pointers are incident on certain scale divisions, then another set of pointers are incident on such and such scale divisions. “Pointers” and “scale divisions” are here used in a generalised sense. The question arises as to how it is possible on the basis of a collection of incidence relations of this sort to build up a quantitative theory i.e. one involving the concept of measurement. It must be noted that until this is done any numbers associated with scale divisions serve merely as labels.

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].