The following algebra possesses certain points of interest and is, I think, worth putting on record; it includes the algebra of matrices as a special case. Consider the algebra H over a ring F defined by

where hpq(p, q = 1, 2, …., n) are linearly independent over F. If are any elements of H, then from (1)

If m, n are integers, m > 0, n > 1, the generalized Stirling numbers are defined by the identity in x,

The infinite integrals

in which λ is positive and – π < θ < π were encountered by Kottler in a problem in the theory of diffraction. They have more recently been studied by Copson and Ferrar, who obtained the remarkably simple Fourier series

in which denotes a “cut Bessel function” of the third kind; this expansion is valid when the term has to be added to the expansion on the right.

The object of this paper is to investigate some properties of series which satisfy conditions of the form

where 0 < ρ ≦ p. denotes, as usual, the n-th Cesàro sum of order p for the series ∑an and the binomial coefficient . It is convenient to state here some properties of and to which we must constantly refer in the sequel.

Schouten and van Kampen (1) have studied the deformation of a . Applying the methods of that paper to the tangent vectors , which exist by hypothesis at all points of a certain region Vm′ (m′ > m) of Vn, we shall have

Whence we define the differentials

In the application of the to the lower index is treated as an ordinal index only. We shall not be concerned with any extension of the to indices other than those of the general Vb (see 1, equ. 3.24).

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.

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.