Professor Hemraj has given a proof of a part of a theorem of Gauss without using the theory of quadratic residues. Proceeding on similar lines, I have obtained a complete proof which is rather simpler and certainly more concise.

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

The object of this paper is to show how the leading properties of Spherical Harmonic Functions may be readily deduced by employing as a typical harmonic a certain simple algebraical expression, which obviously satisfies Laplace's equation; and to extend a similar method to the case of any number of variables.

Various solutions of Mathieu's equation, or the equation of the elliptic cylinder functions

have recently been discussed in an elegant series of papers in these Proceedings. These papers have dealt with the periodic and quasi-periodic solutions, but the present paper merely considers determinants which give the infinite series of relations between a and q, so that the solutions are purely periodic, i.e. the solutions denoted by Professor Whittaker

The present paper is an extension to the general cubic of certain relations connecting two special cubics previously considered. In what follows the term “birational” refers exclusively to the quadric transformation through three fixed points, the correspondence therefore being a whole-plane one for each transformation, and not confined to points on the transformed cubics. Also, the notation is that used in the paper referred to above.

Let X be a topological space. Then we may define the fundamental groupoid πX and also the quotient groupoid (πX)/N for N any wide, totally disconnected, normal subgroupoid N of πX (1). The purpose of this note is to show that if X is locally path-connected and semi-locally 1-connected, then the topology of X determines a “lifted topology” on (πX)/N so that it becomes a topological groupoid over X. With this topology the subspace which is the fibre of the initial point map ∂′: (πX)/N→X over x in X, is the usual covering space of X determined by the normal subgroup N{x} of the fundamental group π(X, x).

The following two propositions are necessary for what is to follow, and are very easy to prove.

I. If A, B, C be three points such that B and C are conjugate with respect to the Polar Conic of A, then are C and A conjugate points with respect to the Polar Conic of B, and similarly with regard to the third vertex.

A triangle possessing the property defined above, viz. that each pair of vertices are conjugate points with respect to the Polar Conic of the third vertex, is called an Apolar Triangle.

In triangle ABC, AD, BE, CF are concurrent at O; through O parallels are drawn to EF, FD, DE, meeting the sides of ABC in L, M, P, Q, S, T, and the sides of DEF in L′, M′, P′, Q′, S′, T′. The two hexagons LMPQST, L′M′P′Q′S′T′ thus formed have the following properties:

(1) The sides L′M′, P′Q′, S′T′ of the latter are parallel to the sides of ABC.

Let $\sOm$ be the closure of a bounded open set in $\mathbb{R}^d$, and, for a sufficiently large integer $\kappa$, let $f\in C^\kappa(\sOm)$ be a real-valued ‘bump’ function, i.e. $\supp(f)\subset\textint(\sOm)$. First, for each $h>0$, we construct a surface spline function $\sigma_h$ whose centres are the vertices of the grid $\mathcal{V}_h=\sOm\cap h\zd$, such that $\sigma_h$ approximates $f$ uniformly over $\sOm$ with the maximal asymptotic accuracy rate for $h\rightarrow0$. Second, if $\ell_1,\ell_2,\dots,\ell_n$ are the Lagrange functions for surface spline interpolation on the grid $\mathcal{V}_h$, we prove that $\max_{x\in\sOm}\sum_{j=1}^n\ell_j^2(x)$ is bounded above independently of the mesh-size $h$. An interesting consequence of these two results for the case of interpolation on $\mathcal{V}_h$ to the values of a bump data function $f$ is obtained by means of the Lebesgue inequality.

AMS 2000 Mathematics subject classification: Primary 41A05; 41A15; 41A25; 41A63

In the paper, “Transformations founded on the Twisted Cubic and its Chord System,” a series of space transformations was described, each of which had the property of transforming the chord system of one cubic into the chord system of the other. In the present paper it is shown that by the aid of a non-singular cubic surface the transformations of orders 1, 2, 3, 4, 5 may be derived directly without the intervention of a space transformation.

It will be found that these transformations form a group which is intimately associated with the 27 lines of the cubic.

A rigid spherical punch vibrates normally on the surface of a semi-infinite isotropic elastic half-space. The essential novelty of this problem, which is treated within the context of classical elasticity, is that of a changing boundary; the radius of the circle of contact on the free surface varies with time. The geometrical co-ordinates are modified to yield a boundary value problem with fixed boundaries. However the governing differential equations become more complicated. These equations are solved by a perturbation procedure for the case where the contact radius a(t) is of the form

where a0 is constant and |ŋ(t)≪1. Finally the normal stress and the total load under the punch are evaluated in the form of series which are valid for sufficiently slowly varying ŋ(t).

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundle

with fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a map

which is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn