The process of randomisation has in recent years come to play such a central part in experimental design that it is of some interest to find that it affords a means of resolving one of the oldest paradoxes which arose in discussions of gaming.

Readers of Todhunter’s The Mathematical Theory of Probability will recall his account (sections 187-190, pp. 106-110) of the correspondence between Montmort and Nicolas Bernoulli on the rule by which the players might guide themselves most advantageously in the game called “le Her”.

The study of the turbulent motion of fluids is of great importance in aerodynamics and engineering, besides being a fundamental problem of hydrodynamics. Several attempts have been made to develop an adequate theory of turbulent motion, but, except for a few good empirical formulae, little advance has been made. L. Prandtl has given a clear and concise account of his own theory and that of Th. von Kármán in a recent paper (1), which contains full references to the literature of the subject. H. Bateman has also contributed an excellent account of the present state of the theory in the Report on Hydrodynamics of the National Research Council of America. In neither of these accounts, however, has reference been made to the theory put forward by J. M. Burgers of Delft, in a series of papers (2) published in 1929, and further developed in 1933. In these papers Burgers has attempted to apply the methods of classical statistical mechanics to the theory of turbulent fluid motion. In this article a simplified account of his methods will be given.

1. Let ABC be a triangle, H its orthocentre, P a point on the circumcircle, and D, E, F the feet of the perpendiculars from P on BC, CA, AB respectively. Then it is known that D, E, F are on a straight line which bisects PH. Through P draw lines making the angle ½π–α in the same sense with PD, PE, PF, PH to meet the sides in D′, E′, F′ and a line through H perpendicular to PH in H′. Then D′, E′, F′ are also on a line which bisects PH′. Call this line the Simson line (α) of P with respect to ABC

It is known that if S be one of a range of conics touching four given lines, and if XP be a variable line through a given point X, then (a) there exists a line p′ which is conjugate to XP with respect to every S, and (b) p′ touches a fixed conic σ, which is the eleven-line conic of X with respect to the range, and which is also the envelope of the polar of X with respect to the range. Let us call this conic the σ of X.