Arago observed in 1811 that if a plane polarised ray of light be passed vertically through a plate of quartz cut at right angles to the crystallographic axis, the ray emerges plane polarised but with its plane of polarisation inclined at an angle to the original plane of polarisation, this angle (or the amount of rotation of the plane of polarisation about the direction of the ray) being proportional to the thickness of the plate. Biot further showed that in some quartz crystals this rotation is in one sense, in others in the opposite. In 1821 Herschel proved that the sense of this rotation was connected with the inclination of the so-called plagiedral faces to the faces of the prism. In 1830 Naumann gave a very complete account of the crystallography of quartz, showed that the two kinds of crystals are mirror images of each other, and gave to this relation the name of Enantiomorphism.

We proved in (1) that every continuous endomorphism can be generated on a subring of the field M. More precisely, the ring H of piecewise polynomial functions has the property that every isomorphism from H into M, continuous in the sequential topology of H, can be extended to a continuous endomorphism of M where the notion of continuity in M is the usual sequential one.

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

The tangent at a point P to a given plane curve intersects another given curve in Q and makes with the tangent at Q to the latter curve a variable angle ψ. It is required to connect the curvatures at P and Q with the length PQ and the angle ψ.

The optical theory referred to in the title of this communication is now fully half a century old; and has, moreover, been well expounded in the standard English treatises of Pendlebury and Heath. Still, notwithstanding its elegance and simplicity, and its great practical importance as giving the first approximation to the theory of the great majority of the optical instruments in ordinary use, its filtration into the strata of popular knowledge has been remarkably slow. It seems, therefore, to be worth while to offer a brief summary of its leading principles, freed as much as possible from the detailed calculations which become necessary when the constants of the optical system have to be deduced from the data of construction, and to indicate methods for experimental verification. In giving this summary, I shall omit the demonstrations of some of the propositions, which can be found by those who desire them in the well known Treatise on Geometrical Optics, by Heath

The contents of this paper were suggested by a discussion of the equation:

in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.

The solutions in series of (1) are:

and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

In an article, published some time since, the author of the present paper deduced an asymptotic expression for the functions of the elliptic cylinder, which expression took the following form:

Here P and Q are certain asymptotic series, and C and a arbitrary constants. General expressions for these constants were not determined in the aforementioned article on account of the difficulties there set forth, though it was pointed out that their numerical calculation for any particular problem was simply a matter of arithmetic computation. It is the object of the present paper to deduce general expressions for these constants C and a in terms of the two parameters which appear in the defining equation for U.

Formulae of interpolation in terms of given central differences might be regarded as falling into two groups, A and B. In group A, the simplest cases are those in which each given difference is one of the two which in the difference table lie nearest to the preceding given difference; the differences are all natural differences (i.e., are not mean differences), and are all expressed in the centraldifference notation. Any such formula can be a central-difference formula for a certain range of the variable: but that is a matter with which we are only incidentally concerned. What I have to do is to examine the formula as determined by the series of differences given. I have then to see how the formula is affected when an ordinary difference is replaced by a mean difference. This brings us to group B, which comprises two formulae only: the Newton-Stirling formula, which expresses the required quantity in terms of a tabulated value and its central differences; and the Newton-Bessel formula, which expresses it in terms of the mean of two tabulated values and the central differences of this mean.

We prove a technical result required by Ivanov and Shpectorov in their construction of a non-split extension of 34371 by the Baby Monster simple group.

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

It is known that if from a given point perpendiculars be let fall on the four Simson lines formed from the four triangles made by taking every, three of four points concyclic with the first, the feet of these perpendiculars lie on a straight line proposed to be called the Simson line of the quadrangle formed by the four points; it is also known that this process can be extended. I propose to examine various results connected with these lines.

The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.

AMS 2000 Mathematics subject classification: Primary 34K05; 34K30

Throughout this paper A denotes an operator function, holomorphic on a deletedneighborhood of a complex number λo, with values in the space ℒ(X,Y) of boundedlinear operators between two complex Banach spaces X and Y. In his survey article(7), I. C. Gohberg has defined for such an arbitrary operator function A the algebraic multiplicity RM(A;λo) and the reduced algebraic multiplicity RM(A;λo) of A at λo. In earlier papers (e.g., (8, 16)) these notions have been defined and studied for morerestricted classes of operator functions. In (8) Gohberg and Sigal treated the case when A is finite-meromorphic at λo, A(λ) is bijective for λ in some deleted neighbor-hood of λo and the constant term A0 in the Laurent expansion of A at λo is aFredholm operator. They proved that in this case