The sequence of orthogonal functions derived from Laguerre-polynomials is known to be complete, and hence closed, in L2 (0, ∞) if (a) > − 1. In a recent paper Dr Kober introduced a generalisation of this sequence, which enables him to extend the known results also for (a) < − 1. Kober's guiding principle seems to be the following one: The Laguerre orthogonal functions form, for (a) > − 1, a complete system of self and skew reciprocal functions of the Hankel transformation of order a. Now, if (a) < − 1, the ordinary Hankel transform has to be replaced by so-called cut Hankel transform. Hence the system of functions which has to replace-Laguerre orthogonal functions when (a) < − 1, should be a complete system of self and skew reciprocal functions of the cut Hankel transformation of order a, such that it reduces for (a) > − 1 (when the cut Hankel transform reduces to the ordinary one) to the sequence of Laguerre orthogonal functions. This, of course, is by no means a unique definition; nevertheless, together with what one would call the permanence of the Mellin transform, it enabled Kober to find a sequence of functions which (i) reduces to the sequence of Laguerre orthogonal functions when (a) > − 1, m = 0, (ii) is a complete set of self and skew reciprocal functions of the cut Hankel transformation with kernel Ja, m and (iii) has the required qualities of completeness and closedness.

The general equation of the second degree in two variables

can be brought by a direct process into the form

the determination of the constants ξ η l, m, n depending only on the solution of quadratic equations; so that the method is suitable for determining the foci, directrices, and eccentricities of conies with given numerical equations.

Recent research on aspects of distributive lattices, p-algebras, double p-algebras and de-Morgan algebras (see [2] and the references therein) has led to the consideration of the classes (n≧1) of distributive lattices having no n + 1-element chain in their poset of prime ideals. In [1] we were obliged to characterize the members of by a sentence in the first-order theory of distributive lattices. Subsequently (see [2]), it was realised that coincides with the class of distributive lattices having n+1-permutable congruences. This result is hereby employed to describe those distributive p-algebras and double p-algebras having n-permutable congruences. As an application, new characterizations of those distributive p-algebras and double p-algebras having the property that their compact congruences are principal are obtained. In addition, those varieties of distributive p-algebras and double p-algebras having n-permutable congruences are announced.

Consider a formal series with partial sums and the corresponding power series . Throughout we will assume that f is analytic for |z| <1, i.e. that A classical theorem of Fatou-Riesz (see (1, 4)) states that if and

then is convergent to 0.

In what follows, character means irreducible complex character.

Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.

In a paper recently printed in the Society's Proceedings, I considered the effect of compressibility in the fluid on the motion of straight vortices; the present paper treats of circular vortex rings in a compressible fluid. The circle passing through the centres of the circular cross sections of the vortex filament will be called the “circular axis,” and the perpendicular to the plane of the circular axis through its centre, the “axis” of the vortex. In the notation employed, a denotes the radius of the circular axis, and e that of the cross section of the filament, while ω represents vorticity, and ρ density. It is also convenient to denote the area of the cross section— i.e., πe, by σ. Following Helmholtz, it will be supposed that e/a is always very small, and that the cross section is truly circular. Certain small inconsistencies in the ordinary theory following from this last assumption will be pointed out, though they do not seem seriously to affect the general applicability of the results. The axis of the vortex ring is taken as axis of z, and z, r, θ are the ordinary cylindrical co-ordinates. It is also convenient to denote by r' the distance of a point from the circular axis of a ring, and by ψ the inclination of this distance to the plane of the circular axis. The effects of the vorticity and variation in density may be considered separately.

1. Points, n in number, A, B, C, D, E, … ., are taken at random in a plane, and through each is drawn a line in a random direction. The only condition imposed is that no two of these lines may be parallel.

(i). Two points A, B, define a circle S (AB) which passes through A, B and the intersection of the random lines through A and B. Its centre is denoted by (AB). Each pair of the points gives such a circle and centre.

A smooth map of a differentiable n-manifold into Euclidean (n+k)-space is called an immersion if its Jacobian has rank n at each point of M. If f is also 1-1, it is called an embedding.

The following note indicates how the equation to the locus of the straight lines which intersect three given lines may be obtained by the use of the conditions that three planes should have a line of intersection.

Three given planes

have a line of intersection if any two of the determinants in the scheme

are equal to zero.

Many-valued or non-Aristotelian calculi of propositions (logics) were originally introduced by generalisation of the truth-table method. It was known by the end of the nineteenth century that ordinary “binary” formulae of the calculus of propositions, such as

could be verified directly by means of the truth-table:

although the terminology and symbolism used were different.

Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):

where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).

In this paper conditions are given that imply that a convolution operator has the same spectrum on all of the spaces Lp(G), 1≦p≦∞.

In this paper we give a method for the solution of the dual integral equations

where Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation for

subject to the usual regularity conditions and the following boundary conditions: