1. It is well known [1] that there is a one-to-one relation between solutions of the Darboux equation and the wave equation. The purpose of this paper is to show that some recent results in the fractional calculus can be used to obtain a similar connection between solutions of Darboux's equation and second order linear hyperbolic differential equations with constant coefficients.

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, if

then

It is probably true that

where ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.

Let R be a ring and let J be the set of all integers. In the set M(R) of all mappings A: J×J→R, let addition and multiplication be defined by

Here aij denotes the image of (i, j) under A and bij, cij, dij are similarly defined for the mappings B, C, D. In (2) we require A, B to be such that the sum is defined and is in R. Thus, in general, M(R) is not closed with respect to multiplication.

A study is made of the length L(h, k) of the continued fraction algorithm for h/k where h and k are co-prime polynomials in a finite field. In addition we investigate the sum of the degrees of the partial quotients in this expansion for h/k, h, k in . The above continued fraction is determined by means of the Euclidean algorithm for the polynomials h, k in .

Of the following properties the first two are obvious; the third was communicated to the Royal Society of London by the Rev. Dr James Booth* in 1854; the others, obtained many years ago, have not as far as I know been remarked. They seem to be more curious than useful.

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

The integration of differentials of the form seems to me to be susceptible of a more methodical mode of treatment than that commonly employed. In the ordinary way of presenting the matter there is little choice left to the student, when such an integration is required of him, between a haphazard, tentative process, and the consultation of a text-book, in which lists of ‘formulae of reduction’ are given.

The condition that the principal normals of one curve may also be the principal normals of a second curve is, as found by Bertrand, that a linear relation with constant coefficients should exist between the curvature and torsion of each curve. In seeking for pairs of curves such that the tangents, principal normals or binomials of one may be the tangents, principal normals or binormals of the other, there are six cases to be considered. The curves of Bertrand are furnished by one case, and a second case, that of evolutes and involutes, is also discussed in the text-books. Of the remaining four only one gives results worthy of mention. Bertrand's problem suggests the inquiry into the nature of the pair of curves when the binormal of one is the principal normal of the other. A certain quadratic relation of a simple character found to exist between the curvature and torsion of the second curve led me to a paper in the Comples Rendus of 1893, by Demoulin, in which the problem had been generalised. His method of solution is different, and no explicit results as to the nature of the curves are given in the paper. Since no indication of the discussion of the problem is given in the text-books I have seen, I venture to submit a note of some results.

The proof given in Chrystal's Algebra, II., pp. 42–5, of this very important theorem is deduced from elementary algebraical principles : and, though somewhat involved, is of great value, as it establishes what must be considered a fundamental theorem in the Calculus.

The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For example

The formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.

The idea in the following note is evidently capable of very wide development, but it can be made clear by a very simple example.

Whatever be the vectors α, β, γ, δ, we have always

But vector operators are to be treated in all respects like vectors, provided each be always kept before its subject.

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is then

where k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.

Théorème.— On considère tous les cercles σ passant par deux points fixes, dont l'un c est sur la circonférence d'un cercle donné s, et l'autre d sur une droite donnèe l. Chacun des cercles σ rencontre la droite l en un second point d' et la circonférence s en un second point c′: La droite c′d' passe par un point fixe i de la circonférence s, quel que soit le cercle σ considéré.

A knot K is said to have tunnel number 1 if there is an embedded arc A in S3, with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K ∪ A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3.