1. If PQRS be an Hyperbola, OE, OF its asymptotes, P, Q, R, S any points on it such that the sectorial area OPQ = sectorial area ORS; and if PA, QB, RC, SD be ordinates to one asymptote and parallel to the other, it is known that

Hence if A, B, C … be taken so that OA, OB, OC … are in continued proportion, the areas OPQ, OQR, ORS … are all equal, and since the number of points can be made as large as we please, the sum of the sectorial areas can be made as large as we please. ∴ The area between an asymptote, the curve, and any radius vector is infinite.

We give a necessary and sufficient condition to determine when an operator in the nest algebra of doubly infinite block upper triangular operators factors through a diagonal projection. An example shows that this condition does not extend to more general nest algebras, but a similar criterion yields a description of the ideals of nest algebras generated by diagonal projections.

The difficulty that is often felt by beginners as to the generality of the proof by projection of the addition theorem in trigonometry and of the formulae for change of axes in coordinate geometry seems to me to be due chiefly to the incomplete treatment of the fundamental theorems on projection. I do not mean that the proofs in the text-books are inaccurate, but only that sufficient pains are not always taken to show how the projection of a directed segment is to be measured so that the generality of subsequent proofs shall be beyond all question.

The convergence of the series is assumed.

The method consists in assuming that the function is equal to a certain power series with undetermined coefficients, substituting these series in the addition formula. This gives an identity.

Picking out coefficient of y we get

1. This paper contains a number of investigations, more or less connected, on the theory of systems of circles. In such a well-worn field one does not expect to have hit upon much that is absolutely new, but it may be hoped that there is sufficient freshness of treatment to give the paper some interest even where it deals with results already known.

Let , (n = 0, 1, ….) be the Laguerre, Hn(x) the Hermite polynomial. Let , be the space of all functions f(x) the pth powers of which are integrable over (a, b), with the norm

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

Most work in genetic algebras has been concerned with inheritance which is symmetric with respect to sex, in that the characters studied are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. When asymmetric situations are studied, the development of the theory is complicated by the higher dimensions of the algebras, and by a feature to which Etherington (3, p. 40) drew attention, namely the fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. Etherington gave some results for the gametic and zygotic algebras of a single sex linked diallelic locus, and its properties were discussed further by Gonshor (4, p. 44). In a second paper (5, p. 334) Gonshor studied sex linkage in the case of multiple alleles, choosing a canonical basis which exhibited very clearly the multiplication table and ideal structure of the algebra. His treatment from the statement of the multiplication table in terms of the natural basis to its expression in terms of a canonical basis, is repeated in the displayed relations (4)–(8) below, for completeness and to establish the present notation.

A common method of solving a linear differential equation consists in expressing the differential operator as a product of factors. The possibility of doing so has been studied extensively by Vessiot, following the work of Picard and Drach, on the lines of the Galois theory of algebraic equations. The analogous process of resolving a, linear differential system, consisting of an equation together with boundary conditions, into two or more systems of lower order does not seem to have been investigated. Such a resolution is not always possible, even in cases where the differential equation can be factorised. Thus the system

is equivalent to the systems

but the system

cannot be so resolved.

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hull

is also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

In this note, we shall prove that the transcendency degree of a finitely generated field extension is equal to a certain integer associated with a restriction mapping of spaces of derivations.

Permit me the liberty of presenting some remarks on an article by Professor Naraniengar inserted in Vol. XXVIII., p. 73, of the Proceedings of the Edinburgh Mathematical Society, 1909–1910.

It is shown that the full automorphism group of a finitely generated group G is virtually free if and only if the center Z(G) is finitely generated of torsion-free rank r at most two and, depending on the value of r, the central quotient G/Z(G) belongs to one of three precisely defined classes of virtually free groups. Some consequences and special cases are also discussed.

Agnew (1) has defined a binary transformation T(α), with α real, as one which takes the sequence {Si}, i = 0, 1,…, into the sequence {si(1,α)} where