Under the auspices of the Edinburgh Mathematical Society, a Mathematical Colloquium is being held in the University Hall, St Andrews. The opening meeting took place last night, and the colloquium will continue in session till July 28.
Courses of lectures are being delivered on "World-Structure by the Kinetic Methods of the Special Theory of Relativity," by Professor E A Milne, Oxford; on "Ramanujan's Note-Books and their Place in Modern Mathematics," by Professor B M Wilson, Dundee; on "Pictorial Geometry," by Professor H W Turnbull, St Andrews; and on "Some Expansions Relating to the Problem of Lattice Points," by Mr W L Ferrar, Fellow of Hertford College, Oxford. In addition to the courses of lectures, there will be single lectures, informal talks, and discussions. Many well-known mathematicians will be present, including Professors W de Sitter, G Temple, and E T Whittaker.
The Colloquium held at St Andrews from July 18 to 28 by the Edinburgh Mathematical Society was the third such meeting held since the War and proved as successful as on previous occasions. Some sixty-five persons attended, including fourteen professors of mathematics drawn from the Universities of England, Scotland, Ireland and Egypt and also Professor W de Sitter, the Director of the Observatory at Leiden and one of the foremost astronomers of the day.
The principal course of lectures, delivered by Professor E A Milne (Oxford), dealt with the problem of world-structure by the kinematical methods of special relativity. This theory, developed by Prof Milne himself, seeks to account for the structure of the system of the spiral nebulae and the recession phenomenon without using general relativity. As fundamental hypotheses it employs, firstly, the idea that events are separated only by time intervals and that "space is a construct of time measurements". Secondly, it lays down a cosmological principle, viz.: that the sequence of an events occurring in the universe and observed by one observer A must be identical with the sequence observed by any other observer B. These observers may be called "fundamental". In addition, the assumption that space-time is Euclidean appears to be made implicitly. On this basis, by means of some very remarkable mathematical analysis, Prof Milne shows how to calculate the density, velocity-distribution and other properties of a system of "particles" each carrying a fundamental observer. These particles are restricted to be in uniform relative motion so that Lorentz transformations hold between their coordinate-systems. But from the same mathematical analysis there emerges the fact that accelerated particles must also occur in the system. Professor Milne calls these "test-particles" and does not consider that they have the same standing in the system as the fundamental particles. Their accelerations, however, he maintains are those which we normally put down to gravitation, and hence he deduces that the constant of gravitation is not a true constant but is varying with the time. In the system of the fundamental particles, the velocity of recession of distant objects is strictly proportional to their distances from the observer.
The physical significance of the particles is two-fold. In the earlier part of the theory they can be identified with spiral nebulae, but in later refinements they are supposed to be some more fundamental type of particle out of which the nuclei of spiral nebulae are formed and which also give rise to cosmic rays. For one of the consequences of the theory is that agglomerations of particles must be formed in time (thus forming the nuclei of the nebulae) whilst there must also be particles which attain enormously high velocities. These, on colliding with slower-moving groups of particles, would give rise to cosmic rays.
In a discussion on these lectures led by Dr W H McCrea (London) in which Prof W. de Sitter and others took part, it was pointed out that Prof Milne's theory had brought into prominence the fact that in general relativity also the scattering of the nebulae went on independently of gravitation. But the exact relation between the two theories had yet to be worked out. It was also not made quite clear in Prof Milne's theory how much of the results depended on the initial assumption that space-time was Euclidean and how much was really independent of the geometry.
In the field of pure mathematics, Professor H W Turnbull (St Andrews) spoke on Pictorial Geometry, dealing with such questions as the generalised construction for the ellipse and hyperbola, the densest and loosest packing of spheres of equal radius within a given volume and the problem of configurations. Mr W L Ferrar (Oxford) gave an account of some expansions relating to the problem of lattice points, treating of lattice points in a circle, the order problem and relations between summation formulae. Of equal interest were the two lectures given by Professor B M Wilson (Dundee) on the notebooks of Ramanujan and the lecture by Professor J M Whittaker (Liverpool) on the representation of integral functions by series of polynomials.
Of the less formal meetings special mention must be made of the two evening discourses. One, delivered by Prof W de Sitter, was a masterly exposition of the subject of the Expanding Universe from the observational point of view and from that of an expert in general relativity. The second, given by Professor G Temple (London), on the General Principles of the Quantum Theory and Eddington's theory of the fine-structure constant, was remarkable not only for the breadth of knowledge the lecturer revealed but also for the fascinating manner in which he presented his subject. Both these lectures aroused interesting discussions. A like interest was displayed in the discussion on Geometry led by Professor J G Semple (Belfast), Dr Timms (St Andrews) and Mr W L Edge (Edinburgh), who, taking a theorem in the theory of three associated quartic curves, each gave a proof of it from a different angle.
The lighter side of the Colloquium was favoured by the good weather and by the situation of University Hall where, by the courtesy of the University Court of St Andrews, the meeting was held and the members housed. At the opening meeting Professor D'Arcy Thompson, on welcoming the members on behalf of the University, spoke also of the history of the town and University. The members were indebted to Professor and Mrs. Turnbull who held a reception to which a number of the residents of St Andrews were also invited. In the afternoons golf was naturally a great attraction, but time was found for two tennis tournaments and an excursion to Loch Earn, which were all much enjoyed by those taking part. Equally successful were two informal concerts organised by the members. The evident keenness of those present for all aspects of the Colloquium encourages the Committee of the Society to believe that these meetings perform a useful function and should be continued in the future.
G C MCVITTIE.
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