Let the set b1b2, b3, …., bk; of k non-negative integers be denoted by Bk. Let ξ;Bk denote the set ξb1 ξb2, ξb3, …. ξbk; ξ being any integer > 1. Without loss of generality, we can suppose that b ≦bi+1.

I have begun to study the laws of the struggle for life by a group of species living in the same environment in such a way that some devour others. I have used for this purpose the so called principle of encounter, considering the encounters of the individuals of the various species, and what follows by reason of the actions that the individuals exercise on one another.

Concerning your paper written in collaboration with Professor Copson, I found the expansion of

for small values of λ in the following manner: The elementary substitution tan ½φ = sin θ/(et + cos θ)yields

i.e. your equation (5.11).

In a general metric space of four dimensions, with an interval given by , where – ds2 = gμνdxμdxν, where g = ║gμν║<0, we can choose locally galilean coordinates at any point. The initial directions of the axes can be fixed in an absolute fashion as the directions of the principal axes of the quadric Gμνdxμdxν = const., is the contracted Riemann-Christoffel tensor.

This note contains an elementary proof of the well-known result that every solution of Besse's equation

with a real parameter n has an infinite number of real positive zeros.

Adopting the notation of Barnes1 and Fox, let us write

If q > p — l, the series on the right of (1) represents an integral function, while if q = p — 1, the series converges only inside or on the circle | x |=1.

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered was

where is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-element

was that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

The elements of the abstract number systems termed groups, rings, ideals, modules and algebras are mere symbols arranged in systems by means of consistent and independent postulates which isolate these systems from the complete realm of abstract mathematics. The postulates are usually chosen so as to generalise the special number systems which have been noticed in traditional mathematics and their independence and consistenc}" are usually proved by means of numerical examples. It is suggested in this note that the extents of the consistency and independence of a set of postulates should also be studied

denote the n-th Cesàro mean of order k for the series aΣan, that is,

where

and let

The object of this note is to deduce from first principles the probability differential of Pearson's x2, and to account for “degrees of freedom.”