This paper contains a summary of some of the results obtained by the author during a study of the propositional part (denoted by A) of a system of free-from-contradiction, type-free logic set up in 1950 by Ackermann (1). It is shown that in its original form the calculus does not possess the desired properties with respect to equivalence of formulae. A calculus A′, which it is shown may be considered as a ‘minimal’ satisfactory extension of A, is constructed. A′ is compared and contrasted with an alternative form A″ of A given by Ackermann in a paper published in 1952 (2). It is proved that A″ is a proper extension of A′. Among the properties of A′ and A″ which are obtained is the resuit that neither calculus possesses a finite complete model. Reference is made to the solution of the decision problem for A, and it is indicated that it is thought that the corresponding problems can probably be solved for A′ and A″. The proofs of many of the results mentioned in the paper are, if given in detail, rather long. In such cases, from space considerations, only outline proofs are given. Complete proofs are contained in (5), which reference also contains several additional properties of the calculi considered.

The method of leading variables is presented as a new method of solving Linear Programming, and is compared with the Simplex and the Dual Simplex methods.

This note describes a simple method of solving the matrix forms of the customary families of finite difference equations which approximate to Poisson's equation (in three dimensions as well as two) and to the bi-harmonic equation. It is intended for use where the boundary of the region over which a solution is wanted is comparatively simple, that is to say, can be subdivided into a comparatively few rectangular areas or rectangular parallelepipeds.

The problem of finding the latent roots and vectors of matrices has been treated in a number of papers ((2)–(7)) mainly from the point of view of desk computers. In this paper the problem is treated from the standpoint of users of high-speed automatic computers. In the first section a number of iterative processes are described and, in the second, the techniques developed for using these processes on the Pilot Model of the Automatic Computing Engine. It is shown that the methods give very high accuracy and can be used to deal with matrices of high orders even on a machine of very limited storage capacity. They have been used on numerous matrices of orders up to 60 mainly on problems arising in the aircraft industry and on eigenvalue problems for systems of ordinary differential equations.

The calculation of the diffraction pattern for an out-of-focus line source viewed through an axially symmetrical lens involves the evaluation of the integral

for a constant value of β (related to the numerical aperture of the lens) and a range of values of x and y. For small values of x and y this can best be done by quadrature; for larger values of x or y or both it is more practicable to use numerical evaluation of an appropriate ordinary differential equation. It is shown that the variation of

with x for y = 0, and with y for x = const., can both be expressed by second-order ordinary differential equations of a convenient form for numerical integration; the evaluation of I1(x, y) from solutions of these equations is straightforward. For comparison with experiment, results are required in the form of I1(x, y) and a derived quantity as functions of y, at relatively widely spaced values of x; numerical work for β = 0.72 has been carried out on the EDSAC and has been organized to produce the results in the required form directly, without interpolation.

Comparison of results obtained by using different Gauss quadrature formulae and different interval lengths show some interesting features of the errors of high-order integration formulae, and show that care is necessary in estimating these errors from the differences between quadratures carried out with intervals of different lengths.

When independent random events are scaled down by a factor n, the number of counts in a given time interval no longer obeys the Poisson Law. The correct distribution function is derived together with its moment-generating function and explicit formulae (n = 2, 4) for the first four moments about the mean. The theory is in agreement with experiments carried out with a scale-of-four counter. A more general treatment is given and is applied to find the variance for all even values of n.

As Armitage(1) has noted, this is a problem of some importance in dust particle counting where overlapping particles are usually counted as single particles owing to their small size and the difficulty of detecting overlapping. In his paper Armitage(1) obtained approximate formulae and also expansions for the expected number of clumps (as he called a self-overlapping group) (i) when the particles were all circular laminae (though of different sizes) and (ii) when the particles were similar rectangular laminae randomly oriented. His expansions were accurate to the first two terms, but he only gave approximate values for the third terms. However, by using a method similar to that of the author (5,6) in treating aggregates in random placings of points and to that of Hammersley (4) in treating one-dimensional problems analogous to those dealt with here, this paper gives exact formulae for any mixture of laminae of different shapes and sizes, the only restrictions being (i) that they are all convex (though they need not be symmetrical), (ii) that they are randomly oriented with no preferred direction, and (iii) that there are a large number of each kind. With these restrictions we shall show in addition that, surprisingly, the only relevant geometrical properties of the laminae are (i) their areas and (ii) their perimeter lengths, their shapes apart from this being immaterial.

Consider a random-walk problem on a simple lattice, the probabilities of the walker taking any direction in the lattice at each lattice point being equal. Then Polya (6) has shown that if a walker starts at the origin and continues to walk indefinitely, the probability of his passing through his starting point is unity in one and two dimensions, but less than unity in three or more dimensions. Recently, a generalization of this problem has been considered (1),(3) in which the walker is allowed to jump several lattice points with assigned probabilities. F. G. Foster and I. J. Good (3) have shown that if the assigned probabilities satisfy certain conditions, Polya's result still holds, and K. L. Chung and W. H. J. Fuchs (1) have shown that the result is valid under far less restrictive conditions. The above authors were primarily concerned with the question whether return is almost certain or not, and did not consider a detailed calculation of the probability at any stage. It is the purpose of the present paper to show that the use of contour integrals allied with the method of steepest descents (4) enables one to perform this calculation very simply.

Sets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.

So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.

The reciprocity theorem for electrical systems which include a radiation link was believed to be true only when the media within the system have dielectric constants which are symmetric tensors. This condition is not fulfilled by the ionosphere, so that the reciprocity theorem is not generally applicable when the radiation link includes one or more reflexions from the ionosphere. It is here proved that, when the ionosphere is horizontally stratified, and when the path from transmitter to receiver is in the magnetic meridian (north-south and south-north transmission), the reciprocity theorem applies (a) when the transmitting and receiving aerials both radiate or receive waves whose electric vector is in the plane of incidence, and (b) when both aerials radiate or receive waves whose electric vector is horizontal. Further, (c) if the electric vector radiated or received is horizontal for one aerial and in the plane of incidence for the other, then there is reciprocity in signal amplitude, but the phase changes for transmission in the two directions differ by 180°. These results are valid for any law of variation of electron density and collision frequency with height. They are based on a ‘full-wave’ theory, and therefore apply to all frequencies. They are unaffected if the path includes multiple reflexions, and if allowance is made for the curvature of the earth.

Two simple laws connecting the amplitude and phase functions of a monochromatic electromagnetic wave of arbitrary form are derived, holding in the case when one of the field vectors is linearly polarized. The first is a generalized Fermat's principle which enables determination of the phase when the amplitude is known; the second expresses the propagation of the (vector) amplitude along the curves orthogonal to the co-phasal surfaces. Some other general properties of linearly polarized fields are also discussed, and illustrative examples are given.

1. An expression in finite terms is found in §§2,3 for the small displacement of a thin elastic plate due to the application of a transverse force at an arbitrary point; the plate is infinite and is bounded internally by an ellipse along which it is clamped. The limiting case in which the ellipse degenerates to a straight line is considered in §4, and the other limiting case of a circular boundary in §5.

In this paper I give a direct proof of a new expansion of the Whittaker function M(a; b; x) as a series of Bessel functions, and some generalizations of this result for the functions 2F2(x) and 4F4(x).

In the course of preparing lectures on elementary quantum theory the author developed a way of presenting time-independent perturbation theory that seems to have some advantages over more familiar presentations. There is nothing fundamentally new in the method, and if any claim of greater power can be made for the new version, it is at best for situations in which high-order perturbations are of interest. Quantum-field theorists will no doubt recognize the relation of this presentation to much that has been done in their subject recently.

If E is a linear set with a positive Lebesgue measure, and E(δ) is its translation through a distance δ, it is obvious that there always exist non-zero values of δ for which E. E(δ) has a positive measure. In the present note I show that for α-dimensional linear sets (0 < α < 1), and α-dimensional measure, the analogous result is not true. This is a further illustration of the deep structural difference between α-dimensional sets for α < 1 and for α = 1.

In an earlier note (3) we have shown by examples that it is possible to have an irreducible non-degenerate point-point correspondence on an algebraic surface such that the correspondence and its inverse have arbitrary unequal negative valencies. We have also remarked that, by multiplying such a correspondence by one with equal positive valencies, we can find correspondences with unequal positive valencies. In (4) we have given reasons, falling short of complete proof, which suggest that, on any surface whose intersection group is cyclic, correspondences with unequal Albanese valencies exist. But the only proofs of the existence of such correspondences are the examples due to the author, Severi and Todd in (3) which all refer to ruled surfaces. Severi has rightly pointed out that these examples are not entirely satisfactory, since, as ruled surfaces have zero geometric genus, there is a confusion between valency correspondences in Severi's and in Albanese's senses, whereas the theoretical arguments of (4) apply only to Albanese valency correspondences. In this note we remark that on hyperelliptic surfaces there exist correspondences with valencies γ and γ3 in the two directions for any integer γ, and that unless |γ| = 1, in which case the two valencies are the same, these correspondences cannot have a Severi valency in either direction. This result is an immediate consequence of some classical results on Abelian varieties (c.f.(2)), but the application seems of sufficient interest to note explicitly. We remark that we are still not able to find examples of correspondences which are irreducible and non-degenerate and whose valencies in the two directions are of opposite sign; it seems a plausible conjecture that such correspondences do not exist.

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.