The Edinburgh Mathematical Society, at the opening of the session on 4th November, 1932, paid tribute to the fame of Professor Sir John Leslie by a Centenary celebration. An account of this will be found in the Edinburgh University Journal for the current year. In view of the recent amazing advances in Science, both pure and applied, it is of interest to observe where mathematics and physics stood at the beginning of this brief interval of exactly one century.

The Diagram shows two complete, and one partially complete, concentric circles. The radii of the largest and smallest are as four to one respectively.

1. The section of certain vegetable cell structures takes normally the form of a net of hexagons. When a departure from this form occurs it is observed to be such that three edges still meet at each vertex of the net and the average number of sides per polygonal mesh is still six. After noting this fact, Graustein shows in a recent paper that under these conditions the average number of sides is necessarily precisely six. He then generalises the geometrical result. The present note derives the same results by an entirely different process. The arguments are of a physical character and do not pretend to be mathematically rigorous. But it seems worth while to put them forward as indicating just why the effect should occur in Nature.

Let Γ(1 + x) = √(2πx)xxe–x φ(x);

This result is Stirling's theorem. A simple proof is given in § 1.87 of Titchmarsh's Theory of Functions (Oxford Univ. Press, 1932).

Rather more than Stirling's theorem can be proved by a method which assumes nothing but the definition of the Γ-function, and Γ (½) = √π, from which it follows that

The formula in question gives the correlation between two variables for a number n1 and nz of individuals when the ordinary parameters, including correlations, are available for the separate groups of n1 and n2 individuals; and it is extended to more than two groups. It is possible, even probable, that the formula is known, but the writer has not found any direct references to it; in any case its practical value is such that there can be no harm in drawing attention to it.

The probability distribution of a bridge hand affords a good example of drawings without replacement from a limited stock.

Let n drawings be made from such a stock. Let prs and qrs be the probabilities of success and failure after there have been r drawings with s successes, and let the probabilities in successive drawings be connected by the relation

1. If there are n individuals A1, A2,.…, An, in how many ways can they be put into groups? For example, if there are three individuals A, B, C, they may be grouped as

that is, in 5 ways, the respective subgroups, 1, 3, 1 in number, corresponding to the partitions 1 + 1 + 1, 1 + 2, 3 of the integer 3. Hence P (3), say, is 5.