The group of the title is one of the classical groups and can be represented—will indeed be represented here—as the group of those projectivities in a certain finite plane that leave invariant the set of zeros of a Hermitian ternary form. Its properties in the large, so to speak, are common knowledge, being particular instances of those of unitary finite groups; the conjugate sets, and sufficient information to determine, or perhaps nearly to determine, the group characters are given in (7). The methods to be used here are elementary; they take advantage of the base field, and consequently the numbers of points in the geometry, being relatively small. Much detailed information emerges that seems worth putting on record; it does not occupy a disproportionate amount of room and might elude the coarser net of an investigation based on generalities.

An extension of Bellman's analogue of Wald's identity is proved for Markov processes with a finite number of states, in which (a) the diagonal matrices of the state functions and the transition matrices may be heterogeneous but are so related that the product of the two matrices which apply at any stage has at least one modal vector in common with all of the corresponding products which apply at other stages, and (b) the probability distribution of the duration of any one realization of the process satisfies certain conditions. The condition (a) is called equimodality. The conditions for (b) are satisfied when there is asymptotically a sufficiently abrupt exponential upper bound to the probability distribution of the duration, and extends simply to validate analogues involving latent roots other than the one which appears in Bellman's formula. A certain generalization of Wald's stopping rule produces a bound of this kind.

Andrews and Curtis have shown (1) that one can embed two Sn's in En+2 for n = 2, in such a way that one sphere cannot be shrunk to a point in the residue space of the other. In this paper the result is shown to be true for any n ≥ 1. (The result is obvious for n = 1.) The method is to calculate the appropriate homotopy group of the residue space of one sphere, and to show that the embedding of the other sphere represents a non-zero element of the group. The two spheres can both be embedded analytically.

If {λ;n}, {bn} are sequences of complex numbers, and we consider the series ∑bn exp (−λnx), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x)∈L, then, writing

(if λn = 0 the corresponding term is ½bnx2) where the series is supposed is to be uniformly convergent in (0, 1), we have

for 0<h<h(x).If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence {φv(x)} of integrable functions with the property that

then, on multiplying (1) by φv(x) and integrating over (0,1), we obtain a formula for bv in terms of F(x). On integrating by parts twice, bv will be expressed in terms of f(x), and this will constitute a uniqueness theorem for the series ∑bn exp (−λnx).

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.

Although there are many variations of finite-difference methods of obtaining approximate numerical solutions to ordinary differential equations they share the common feature that they tend to treat an equation of a given type as a standard problem and take no account of any special characteristics the wanted solution may have. We here suggest an alternative procedure when the wanted solution exhibits exponential characteristics. In essence the idea is that if a differential equation has an exponential type solution y(x) it is useful to solve numerically, instead of the equation for y, the equation for u = logey. The error-building and stability characteristics are then those of u rather than y and consequently the accuracy of the solutions may be improved. Although there is nothing basically new in this, of course, the point that we demonstrate is that the differential equation in y can be solved numerically in such a manner that the transformation from y to u is not actually carried out, i.e. we retain the original dependent variable but take account of the exponential variation by modifying the integration formula. Consider for example, in the usual notation, the first-order equation

with a given initial condition y(x0) = y0. If x0, x1, …, xr, xn is a set of pivotal values of x;, usually assumed equally spaced so that xr+1 − xr = h, the usual approach replaces (1) by the formula

which, once the integral is expressed in terms of pivotal values of f using a difference series, represents a step-by-step formula for constructing successive values of y.

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformation

where is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed as

such that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).

A recently developed theory of the electromagnetic field in nearly periodic structures is used to calculate the field in an Alvarez type linear accelerator. The problem is reduced to the solution of the transmission line equations for the ‘voltage’ and ‘current’ of the dominant mode. Formulae are given for the coefficients in these equations, and also for the ratio of mode current to accelerating field, as functions of the cavity dimensions of the accelerator; numerical values are given for the Berkeley accelerator.

Two-dimensional elastic waves in a half-space 0 ≤ r < ∞, 0 ≤ θ ≤ π are examined under the assumption of dynamic similarity, so that the stresses depend only on r/t, θ. Analytic solutions are given for constant surface traction on θ = 0, 0 < r/t < V, where V is constant, the rest of the surface being unloaded, and for a concentrated load at r = 0.

Numerical results are quoted for the particular case V → ∞, corresponding to a load on half the bounding plane.

The notion of the order of convergence of iterative processes may well have been known to Gauss or even Newton, but to the author's knowledge it was first considered in detail in 1870 by Schröder(5). The subject was later investigated independently by Hartree(4) and Bodewig(1). Hartree alone mentions, briefly, the extension from one to several variables, but gives no detailed analysis. Domb and Fisher(2) consider a particular type of iterative process in several variables, which they call ‘degenerate’, in which all the variables tend to the same value. We shall follow essentially Hartree's treatment, in considering the solution of r simultaneous equations in r variables and applications to a generalization of the Newton-Raphson process.