It is shown that all the classical thermodynamical properties and the equation of state of a rectilinear assembly of spherical molecules with short-range attractive forces acting only between neighbours can be derived from a single partition function which is an analytic function of the pressure and the temperature. The thermodynamical functions are simply related to the Laplace transform of the function e−E(x)/(kT), where E(x) is the interaction energy between two neighbouring molecules. Although this linear model behaves like a perfect gas at high temperatures and like a crystal near the absolute zero, there are no critical phenomena, owing to the analytic character of the partition function. It is, however, pointed out that at very low temperatures the slope of the isothermal curves can suffer extremely sharp changes which may be interpreted as changes of phase, and this point is illustrated by an example.

Finally, I should like to express my sincere gratitude to Prof. H. Jones for his guidance of this work and for his invaluable advice throughout. My thanks are also due to the Turkish Ministry of Education for the award of a scholarship.

The history of mechanical transport has been one of increasing speeds and there seems to be little doubt that air transport will follow this pattern. To the passenger, increased speeds mean a saving of time, and time is money. To the operator, increased speeds mean more flights per week and therefore greater earning capacity for his aircraft. This year indeed has been marked by the appearance of the four-jet air liner, the forerunner of the four hundred knot air transport.

The results of this paper have as their background two concepts fundamental in the theory of non-associative algebras (i.e. algebras in which the associative law of multiplication is not assumed). These concepts are:

(1) Isotopy,

(2)The three nuclei λ, μ, ρ of an algebra .

This paper deals with the improvement to be gained by using position lines from two Decca chains to obtain a fix in areas where a single chain fix is liable to extensive errors.

Recently The Decca Navigator Company, with the cooperation of the Admiralty, undertook a series of trials between Scotland and southern Norway, using the English and Danish chains of stations outside the normally used coverage of either chain. The results obtained were highly satisfactory and an account of them is given in this paper.

1. Any mention of the word ‘grouping’ immediately brings to a statistician's mind the Sheppard corrections. These are usually used to make inferences about the underlying ungrouped population from observations made on the grouped population, but it is important to realize that, as stated and proved, they have nothing to do with sampling or inference and are merely expressions for the moments of one population in terms of the moments of another population derived from it. They can only be used for the inference problem when allied to the method of moments. This method, as formulated by K. Pearson, consists in taking for θ* the estimate of the population parameter θ, the same function of the sample moments mi that θ is of the population moments μi, each mi being an estimate of the corresponding μi. If the population is grouped the mi are estimates of the , the grouped population moments, so we require θ as a function of the to apply Pearson's method. This can be done since θ is known as a function of the µi and the µi are known as functions of the by the corrections. Use of the Sheppard corrections with any other inference method, even when this method, when applied to the continuous population, yields an estimate which is a sample moment, so far as I am aware, has not been examined except for the normal curve.

1. The following sentence occurs in a recent paper ((1), p. 54): we do not understand precisely what it means, but we shall show that the idea behind it can be put on a reasonably quantitative basis: ‘In the vocabulary of solvers of simultaneous equations, the operator (1, −2, 1) tends to lead to ill-conditioning when the number of equations is large.’

1. It is frequently required to find the numerical value of the definite integral It is, however, often found that even if the analytical expression of f(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases where f(x) is not even defined by an analytical expression but merely specified by a table of numerical values. In the two cases mentioned one has to resort to a numerical integration procedure in which the integral is evaluated in terms of the values of f(x) at a finite number of the arguments x1, x2, …, xn. There are essentially two different types of quadrature formulae available, those which depend on the use of finite differences and the tabulation of f(x) at equidistant points, and those which depend on the values of f(x) at suitably selected points xi. These latter, so called mean-value methods, will be discussed in the present paper. For a recent treatment of such methods see, for example, Beard (5).

As between the scholar and the technician, the armchair theorist and the practical man, the antagonism is as old as civilization itself. In Queen Elizabeth's days the gentleman born was the only gentleman, however ignorant, while learning was the preserve of the universities; the practical man was a mere ‘mechanician’. But there was some liberal thinking, common-sense thinking, even in those days, and one Gabriel Harvey, a well-known literary man and controversialist, spoke up boldly for the technicians. Even though they had heard no university lectures, he said, and were deficient in book-learning, it would be a bold man who would despise such expert practitioners as Humfrey Cole, the mathematical instrument maker, William Bourne, the naval gunner, Matthew Baker, the shipwright, John Shute, the architect, John Hester, the chemist, and Robert Norman, the hydrographer and compass maker. No fewer than four out of the six notable men he named were doing work that was relevant to the advancement of the navy and navigation. They were at the height of their powers in the decade preceding the defeat of the Spanish Armada, a coincidence which cannot be without significance.

The effect of radar on the application of the Rule of the Road at Sea has been given attention in several papers and discussions in recent months. Deductions on their relationship have been attempted, and already judgements have been delivered in the courts of this and other countries which take account of the installation of radar in one or both of the parties to a collision. The suggestion has been seen that guidance to mariners concerning their actions and responsibilities when radar is fitted must await the development of case law. This development will be a lengthy and probably painful process and the object of this paper is to discuss the possibility of establishing principles which would enable the great potential value and legitimate application of radar to be established without waiting for casualties.

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:

Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:

(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differences

with respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;

if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfying

is measurable for any realbi, tiand has measure F(t; b).

In 1914 Mitchell (13) underook the determination of the finite primitive collineation groups in more than three dimensions which are generated by homologies. Among these groups are five which are remarkable as not being members of an infinitely extended series, and these five isolated groups fall naturally into two sets. One set comprises three groups of orders 27.34:5, 29.34.5.7 and 213.35.52.7, belonging respectively to space of five, six and seven dimensions, the centres of the generating homologies of the smaller groups being among those of the larger ones; these groups had been considered earlier by Burnside (3) as groups of linear substitutions with rational coefficients, and both the groups and the configurations formed by the centres of the generating homologies are closely related to polytopes in Euclidean space of six, seven and eight dimensions first described by Gosset (9) and later studied in detail by Coxeter (6, 7). The two remaining groups, of orders 26.34.5 and 28.36.5.7, belong respectively to four and five dimensions, and the centres of the forty-five generating homologies of the former appear among the centres of the 126 generating homologies of the latter. The smaller group is the well-known simple group of order 25920, and is probably the most extensively studied of all special groups of finite order. Its representation as a collineation group in four dimensions was first obtained by Burkhardt (2), who determined the invariants of the group; the simplest of these is a quartic primal with nodes at the 45 centres of homologies, which was later studied by Coble (5) and myself (14). The geometry of the configuration formed by these 45 nodes has been investigated in detail by Baker (1), whose results I have used elsewhere (15) to classify the operations of the group and obtain their distribution into conjugate sets.