The area of the pedal triangle of a given triangle is easily shown by trilinear co-ordinates to bear to that of the original triangle the ratio R2 – S2: 4R2 where S is the distance of the point from the circumcentre of the triangle. A proof, by purely geometrical methods, of this theorem was read before the Society (Proceedings, Vol. III., pp. 78–79) by Mr Alison.

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.

Riesz potentials with radially symmetric densities are examined from the standpoint of Mellin multipliers. Various results are deduced from the underlying multipliers, including a decomposition of the potential into a product of Erdélyi-Kober fractional integrals. Distributional versions of these results are also produced and shown to be valid under less severe restrictions on the parameters than those required in a weighted Lp setting.

1. In the Cambridge and Dublin Mathematical Journal, vol. v., 1859, De Morgan gives the definition of the “area contained within a circuit” as the area swept out by a radius vector which has one end (the pole) fixed and the other describing the circuit (in a determinate mode), on the supposition that each element of area is positive or negative, according as the radius is revolving positively or negatively. He remarks that the definition satisfies existing notions, that it provides the necessary extension of the meaning of the word area, and proceeds to show that it gives to every circuit the same area, whatever point the pole may be. The object of this paper is to give an Area-Theory beginning with the triangle and going on to circuits bounded by straight or curved lines. The fundamental proposition is derived from Analysis, and the geometry of the applications is therefore an Analytical Geometry; indeed, one of the objects of the paper is to emphasise the advantage of keeping Analysis and Geometry in close correspondence. As evidence of the difficulty of pursuing an Area-Theory in Geometry, without the aid of Analysis, it may be noticed that Townsend in his Modern Geometry (1863), §83, lays down Salmon's Theorem in this form: “If A, B, C, D be any four points on a circle taken in the order of their disposition, and P any fifth point, without, within, or upon the circle, but not at infinity, then always area BCD.AP2 − area CDA.BP2 + area DAB.CP2 − area ABC.DP2 = 0, regard being had only to the absolute magnitudes of the several areas which from their disposition are incapable of being compared in sign.”

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian

(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:

where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.

Let E be an injective module over the commutative Noetherian ring A, and let a be an ideal of A. The A-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈N is ultimately constant. This result is analogous to a theorem of M. Brodmann that, if M is a finitely generated A-module, then the sequence of sets (AssA(M/αnM))n∈N is ultimately constant.

The usual definition of hyperbolicity of a group G demands that all geodesic triangles in the Cayley graph of G should be thin. Using the theorem that a susbquadratic isoperimetric inequality implies a linear one, we show that it is in fact only necessary for all triangles from a given combing to be thin, thus giving a new criterion for hyperbolicity of finitely presented groups.

It is well known that in a given triangle a one-fold infinity of triangles may be inscribed similar to a given triangle This becomes at once obvious on consideration of the converse problem; for we may circumscribe about a given triangle (A), a triangle similar to a second triangle (B), and having its sides parallel to the sides of (B).

A simple approximate formula is obtained for the capacity of the condenser formed by a “small” conductor placed inside a much larger one. The formula involves a constant whose choice is, to a certain extent, arbitrary and it is shown that, for problems involving spheroids inside cylinders and between parallel plates, the constant may be found in a simple fashion so as to give very accurate results. A similar formula is obtained for the loss in potential energy due to a crack or cavity in a circular beam or a thick plate. For the particular cases of the boundary value problems considered which have been treated by other means very close numerical agreement is obtained between those results and ones deduced in the present paper.

Following (1) we say that a subgroup H of a group G is almost subnormal in G if H is of finite index in some subnormal subgroup of G, or, equivalently, if |Hn : H| is finite for some n, where Hn is the n-th term of the normal closure series of H in G. The aim of this article is to prove, in answer to a question of R. Baer, the following analogue of the well known result of Roseblade and Stonehewer (3) that in any group the join of a pair of finitely generated subnormal subgroups is always subnormal:

The object of this paper is to show how some formulae in Analytic Number Theory, in particular, the formula for N(T), the number of zeros of the Zeta-function between t = 0 and t = T, are easy deductions from the Generalised Poisson-Jensen formula. A similar method, using Green's function instead of the general function g(s) of § 2, has been published by F. and R. Nevanlinna (Math. Zeitschrift, 20 (1924), and 23 (1925), but the result contained in (vi) below appears to be new, although the writer has not been able, as yet, to make any effective use of it. It is clear that other applications could be made, but it seems sufficient to give here an indication of the method. The notation throughout is the usual one, and the references are to the Cambridge Tract by E. C. Titchmarsh on “The Zeta-function of Riemann”. Finally, I am indebted to the referee for the reference to the papers of F. and R. Nevanlinna.

We show that every continuous nest of bounded multiplicity is unitarily equivalent to itself in a non-trivial way. Along the way, it is shown that no finite (measurable) partition of the unit interval can separate absolutely continuous homeomorphisms.

The theorem of Heilermann can be stated thus:—

If the series

is converted into a continued fraction of the form

then the elements of the continued fraction are

where

and is obtained from this determinant by deleting the (r + 1)th column and the last row.

Consider the nonautonomous delay logistic difference equation

where (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.