1. Summary. In Biometrika (1948) a paper [1] by H. L. Seal contained a theorem applying to “n random variables normally distributed about zero mean with unit variance, these variables being connected by means of k linear relations.” Arising from this is the question of how to obtain a set of normal variates connected by k linear relations and such that each variate has unit variance: or, more generally, connected by k linear relations and such that each variate has a given variance.

In 1913 L. Aubry [1] proved the following theorem:

If a and m are relatively prime, m > 0, and if is not an integer, then it is always possible to find integers x and y not both zero such that

Though much effort has been expended in studying the mean values of arithmetic functions there is one case which has not yielded a great deal either to elementary or analytic methods. The case to which we refer is that of estimating

A summation method defined by the linear transformation

will be called an l-l method if ∑|yr| < ∞ whenever ∑|xk| < ∞; if in addition we have ∑yr = ∑xk whenever ∑|xk| < ∞ we shall say the method is absolutely regular. (It should be observed that we are dealing with series-to-series methods, not sequence-to-sequence as usual.)

The two ternary quadratic forms

are said to be reciprocal (to each other) since their coefficients form inverse matrices :

Synopsis. This is the first of two papers dealing with the projective theory of spinors. It contains the algebraic introduction to the projective spinor analysis which will be dealt with in the second paper.

The law of transformation for a general mixed tensor

1

is somewhat complicated at first sight, but algebraically it can be stated succinctly in the following manner. If, without regard to the original position of various indices, we denote the components of the tensor in some fixed simple ordering by

2

This paper consists of two parts. The first contains an outline of the theorems and principal results and the second (§§2-6) gives proofs of the theorems and additional details. The theorems concern properties of Darboux continuous functions and functions having generalized Darboux properties. The corresponding results are shown to have interesting applications to the theory of non-absolutely convergent integrals.

Let g denote an odd prime, and h = h(g) the class number of the cyclotomic field R(), where is a primitive gth root of unity. It is known that we can write

Let it be required to arrange v elements into v sets such that each set contains exactly k distinct elements and such that each pair of sets has exactly λ elements in common (0 < λ < k < v). This problem we refer to as the v, k,λ combinatorial problem.

Cet article supposera connue la théorie des distributions, pour laquelle nous renverrons à notre livre Théorie des distributions (Paris, 1950). Nous noterons par (D) cet ouvrage. Rappelons cependant les définitions des principaux espaces utilisés, sur une variété indéfiniment différentiable Vn: