Let μ ≡ refractive index of the prism.

Let α, α′, β′, β be the successive angles of incidence and refraction of a ray, in a plane perpendicular to the edge of the prism.

This account of our Society is based to some extent on my Presidential address, which was given on 19 October 1977 and was devoted to the first fifty years.

In the latter half of the nineteenth century there was an upsurge of interest in mathematics that resulted in the foundation of a number of mathematical societies in different countries. The London Mathematical Society (1865), the Moscow Mathematical Society (1867), the Société Mathématique de France (1873), the Edinburgh Mathematical Society (1883) and the New York (later American) Mathematical Society (1888) were all founded in this period. There had, of course, been earlier more local societies, such as the Spittalfields Mathematical Society, which flourished over a long period before becoming defunct, as well as one or two much older bodies, for example the Mathematische Gesellschaft in Hamburg (1690), which still survive.

The theory of Inversion presents one of the simplest examples of those Birational Transformations of plane figures, whose general theory is due to Cremona. It has a distinguishing feature to which it owes its name. If the point P “inverts” into Q, then Q inverts into P. It is therefore a simple case of these involutive point transformations much of the general theory of which was developed by the late Admiral de Jonquières in a paper printed as late as 1864 in the Nouvelles Annales, but which had originally been addressed to the Institute of France in 1859. This memoir is not only highly interesting, but is eminently readable and very ingenious.

In what follows, a triangle ABC is taken, from it another triangle A′B′C′ is formed so that

A is the mid point of B′C′,

B is the mid point of C′A′,

and C is the mid point of A′B′.

Erdős (1) has proved the following result:

Theorem A.Every integral polynomial g(n) of degree k ≧ 3, represents for infinitely many integers n a(k-1)th power-free integer provided, in the case where k is a power of 2, there exists an integer n such that g(n)≢0 (mod 2k-1).

If ρ be the vector of a corner of a square in one system, σ that in a system derived without inversion, we must obviously have

k being the unit-vector perpendicular to the common plane.

August Ferdinand Möbius was born at Schulpforta, in Saxony, in the year 1790. He studied in the Universities of Leipsic and Göttingen, and, at the age of 2G, was appointed extraordinary Professor of Astronomy and Superintendent of the Observatory at Leipsic. There he remained till his death, in 1868, being appointed ordinary professor in 1844. Between the years 1817 and 1868 Möbius wrote his Barycentric Calculus, a Treatise on Statics, another on the Mechanics of the Heavens, and a large number of papers on Mathematical, Dynamical, and Astronomical questions. Most of these papers were contributed to Crelle's Journal, which was founded in 1826. The works of Mobius have recently been collected under the direction of the Royal Scientific Society of Leipsic, and under the editorship of Klein, Scheibner, and Baltzer.

We answer a question raised in an earlier paper concerning the pure state space of a separable C*-algebra.

A theory of positive definite kernels in the context of Hilbert C*-modules is presented. Applications are given, including a representation of a Hilbert C*-module as a concrete space of operators and a construction of the exterior tensor product of two Hilbert C*-modules.

The possibility of the steady motion of a spherical vortex of constant vorticity in an infinite homogeneous liquid was first pointed out by Hill in the Phil. Trans., 1894, pp. 213–245. He had already discussed a case of motion which had for the surfaces always containing the same particles those given by the equation

a particular surface being

It is known, for each 1<p<∞, p≠2, that there exist differential operators in LP(ℝN) which are not (unbounded) decomposable operators in the sense of C. Foiaş. In this note we exhibit large classes of differential (and unbounded multiplier operators which are decomposable in LP(ℝN) and hence have good spectral mapping properties; the arguments are based on the existence of a sufficiently rich functional calculus. The basic idea is to take advantage of existing classical results on p-multipliers and use them to generate appropriate functional calculi.

It is shown that if G is a group and Aut G is a countable periodic CC-group then Aut G is FC.

Minkowski proved the following (for a proof see (4)): if A and B are n × n positive semi-definite hermitian matrices then

It is known (4) that if both A and B are non-singular, then the equality holds in (1) if and only if B = cA where c is a positive number.

This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,

For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li by

is again in L1, and

We determine all functions f(z) meromorphic in the plane such that f′(z)/f(z) has finite order and f(z) and F(z) have only finitely many zeros, where F(z) = f″(z) + Af(z) for some constant A.