A lemma used in the above paper can be extended to cover the case of a sequence Sn determined by a recurrence relation of the form

r being any positive integer.

For convenience denote everywhere. Let us suppose that an inequality of the form

holds for p = 2, 3, … ., n — 1. The constant K can be determined from the initial values of Sn, when kn and ln have been found.

§ 1 Introduction. The formulae

where n, q, m are positive integers, are well known. I am not aware, however, of any paper dealing with the corresponding definite integrals of products of two Ferrers' Associated Legendre Functions of different integral orders and degrees. With the help of the formulae given in my previous paper in this volume it is possible to discuss these integrals. I have found in the case of the first integral that if the orders of the two associated functions are both odd or both even and if, in each case, the degrees are both odd or both even or equal, the integral has usually a non-zero value. If, however, the orders are both odd or both even and if the degree of the function having the greater order is less than that of the other, or if the degrees are one odd and one even, the integral vanishes. Further, if one of the orders be odd and the other even, and the corresponding degrees are also even and odd (or odd and even), the integral has a non-zero value; in all other cases it vanishes. The values of the second integral can be obtained in a similar manner. An example of the methods employed and the results obtained is given in the following section.

The purposes of the following note are these:—(1) To show the relation between Whittaker's Cardinal Function and Fourier's Repeated Integral; (2) to give a new derivation of Fourier's Integral Formula; and (3) to extend the notion of the Fourier Integral to the case in which the variables involved are complex.

B. Pascal and others have studied the properties of the table:

Take the point P in the arc AB, and let ∠ ABP = θ, then the trilinear equation of the Wallace line of P (p say) is

(a) The lines AP, BQ, CR drawn parallel to the sides BC, CA, AB of the triangle ABC.

In this case p, q, r, by [§ 17] are readily, seen to be concurrent.

Let A denote a complex Banach algebra with unit, Inv(A) the set of invertible elements of A, Sp(a) and r(a) the spectrum and spectral radius respectively of an element a of A. Let Γ denote the set of elements of A whose spectra contain non-negative real numbers, i.e.

For positive integral n and positive real x1, … xn let

Where

and let

In recent years the geometry of generic submanifolds of Euclidean space has been theobject of much study. Thorn hinted in [7] that the focal set of such a submanifold couldprofitably be studied by using the family of distance squared functions on thesubmanifold from points of the ambient space. For a generic submanifold the focal set isthe catastrophe or bifurcation set of this family. The key to obtaining results on thelocal structure of this focal set is a transversality theorem of Looijenga [5]; for analternative exposition see [8].

The perpendicular from A on BC = AB·AC/2R, which is constant.

In (12) we introduced the concept of essential separability and used it to define two classes of locally convex spaces, δ-barrelled spaces and infra-δ-spaces, which serve as domain and range spaces respectively in certain closed graph theorems (12, Theorems 3 and 7). In this note we continue the study of these ideas. The relevant definitions are reproduced below.

Some axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.

The classical von Neumann–Oxtoby–Ulam Theorem states the following:

Given non-atomic Borel probability measures μ, λ on In such that

there exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set S

It is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

The object of this note is to show that, in applying the methods of hypercomplex numbers to the theory of determinants, there is, for many purposes, no gain in using a particular number system.

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

Let S be the focus of the conic, PX the directrix, e the eccentricity.

Let V be the middle point of PP′.

Draw VK perpendicular to the directrix, and with centre V describe a circle, radius equal to e.VK.

Join SP, SP′ and draw radii Vp, Vp′, parallel to SP′, SP, and let PP′ meet the directrix in L. Then p, p′ are on the line SL.

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).