Harold Jeffreys

Abstract Jeffreys' Theory of Probability, first published in 1939, was the first attempt to develop a fundamental theory of scientific inference based on Bayesian statistics. His ideas were well ahead of their time and it is only in the past ten years that the subject of Bayes' factors has been significantly developed and extended. Recent work has made Bayesian statistics an essential subject for graduate students and researchers. This seminal book is their starting point.

Abstract It is shown that a certain differential form depending on the values of the parameters in a law of chance is invariant for all transformations of the parameters when the law is differentiable with regard to all parameters. For laws containing a location and a scale parameter a form with a somewhat restricted type of invariance is found even when the law is not everywhere differentiable with regard to the parameters. This form has the properties required to give a general rule for stating the prior probability in a large class of estimation problems.

DR. KEYNES'S book is a searching analysis of the fundamental principles of the theory of probability and of the particular judgments involved in its application to concrete problems. He adopts the view that knowledge may be relevant to our rational belief of a proposition without amounting to complete proof or disproof of it, and treats the probability as a measure of this relevance. NO.Otherwise he does not attempt to define "probability," regarding it as a concept intelligible without further definition. In this respect, as in several others, he is in agreement with the views expressed by Dr. Wrinch and the present reviewer (Philosophical Magazine, vol. 38, 1919, pp. 715-31), and some comparison of the two presentations may not be out of place. A Treatise on Probability By J. M. Keynes. Pp. xi + 466. (London: Macmillan and Co., Ltd., 1921.) 18s. net.

This well-known text and reference contains an account of those parts of mathematics that are most frequently needed in physics. As a working rule, it includes methods which have applications in at least two branches of physics. The authors have aimed at a high standard of rigour and have not accepted the often-quoted opinion that 'any argument is good enough if it is intended to be used by scientists'. At the same time, they have not attempted to achieve greater generality than is required for the physical applications: this often leads to considerable simplification of the mathematics. Particular attention is also paid to the conditions under which theorems hold. Examples of the practical use of the methods developed are given in the text: these are taken from a wide range of physics, including dynamics, hydrodynamics, elasticity, electromagnetism, heat conduction, wave motion and quantum theory. Exercises accompany each chapter.

Abstract It is well known that in certain circumstances a type of instability may arise at the surface of separation of two fluids when there is a finite difference between the velocities on the two sides of the surface. Some disturbances of the surface, of simple harmonic type, may increase exponentially in amplitude until the customary simplifying assumption, that the terms of the second degree in the displacements from the undisturbed state can be ignored, breaks down. One would naturally expect that in the case, for instance, of a wind blowing over the surface of water, waves would be first formed when the velocity of the wind is just great enough to make one particular type of wave grow; thus the critical wind velocity and the wave-length of the waves first formed will constitute checks on any theory of wave formation. The problem for frictionless fluids has been solved by Lord Kelvin, subject to the restriction that the disturbances considered are two-dimensional, no horizontal displacement occurring across the relative velocity of the fluids. Since, however, the possible initial deformations of a horizontal surface will not as a rule satisfy this condition, an investigation of the growth or decay of deformations of other types is desirable. 1. Hypothesis of Irrotational Motion. Let the two fluids be incompressible (a legitimate approximation so long as the wave velocity is small compared with that of sound in either fluid) and of great vertical extent. Let the origin be in the undisturbed position of the surface of separation and the axis of z vertically upwards. Let ζ be the elevation of the surface, and suppose the two fluids to have initially velocities U and U' parallel to the axis of x, accents referring to the upper fluid. Let the densities of the fluids be respectively ρ and ρ', and the velocity potentials in them Ф and Ф'. Let the operators ∂/∂t, ∂/∂x, ∂/∂y, ∂/∂z be denoted by σ, p, q, and ϑ respectively. Putting r2 for — (p2 + q2 we see that ∇2Φ = 0 (1) is equivalent to (ϑ2 — r2)Ф = 0, (2) whence Ф = Ux + erz A, (3) where A is a function of x and y, determined by the value of Ф where z is zero.