A short proof of Chow's generalization of Widder's inequality and generalizations of some related Hardy's results are given.

The differential equation in question is of the second order and has three regular singular points. It is usually denoted by

where a, b, c are the singular points, a, a′ etc. the exponents at those singularities. The solution when no one of the numbers α – α′, β – β′, γ – γ′ is an integer or zero is well known; all types of solution are expressible in terms of the hypergeometric functions.

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let π: U → N be a horizontally homothetic harmonic morphism. In this paper we show that if π has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometrics of M and N one of six well known examples.

The following arrangement of the proof of this theorem could, I think, be given at a comparatively early stage, even if the necessary case of De Moivre's theorem had to be proved as an introductory lemma.

Let u, v be two rational integral algebraic functions of x, y with real coefficients, and let c be a simple closed contour in the plane. As the point (x, y) travels round c let those changes in the sign of u that take place when v is positive be marked and let (u, v; c) denote the excess in number among these of changes from + to − over changes from − to + *.

Let N be an arbitrary near-ring. Each element a ∈ N determines in a natural way a new multiplication on the elements of N which results in a near-ring Na whose additive group coincides with that of N but whose multiplicative semigroup generally differs. Specifically, we define the product x * y of two elements in Na by x * y = x a y where a product in the original near-ring is denoted by juxtaposition. One easily checks that Na is a near-ring with addition identical to that of N. The original near-ring N will be referred to as the base near-ring, Na will be referred to as a laminated near-ring of N and a will be referred to as the laminating element or sometimes more simply as the laminator.

It can be shown as follows that Newton's Theorem can be derived from elementary considerations without making use of the idea of an equation and its roots.

The equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

The subject of the Singular Solutions of Differential Equations of higher orders than the first is not touched in the ordinary textbooks. Their existence, for instance, is not mentioned by Forsyth in his Treatise. This is probably due to the fact that, while in the case of equations of the first order a theory has been developed by Cayley and others which connects the singular solution in a geometrical manner with the ordinary solutions (the singular solution being, of course, the envelope of the ordinary solutions), in the case of equations of, say, the second order no corresponding theory exists—at any rate, no corresponding theory has yet been developed. Our only guide in the subject at present is Cauchy's Existence Theorem, which points out where we are to look for singular solutions.

In 1640, when only 18 years of age, Pascal published a tract of a few pages with the above title. It contains only a few enunciations, and concludes with the statement that the author has several other theorems and problems, but that his inexperience, and the distrust he has of his own powers, do not allow him to publish them till they have been examined by competent judges. He afterwards wrote a complete work (opus completum) on the Conics, which was submitted to Leibnitz by M. Périer, Pascal's brother-in-law. Leibnitz recommended that it should be published; but this was not done, and we know its contents only from the analysis which Leibnitz sent back to M. Périer.