In this paper the equations of the meson are treated in the same manner as the Dirac equation in a previous paper (K. Fuchs, 1940, Part IV). We use the formulae developed by Kemmer (1939), with a small modification, introduced so that the set of Maxwell's equations for the electro-magnetic field is obtained as a special case.

9. A formula is adopted which gives the probability of an individual of given ability passing a test item in terms of two quantities constant for that item. A method of estimating these two constants is given. Making certain assumptions concerning the items composing a test, formula? are then derived giving the expected value and the standard error of the test score of any person. It is shown that there is a certain reciprocity between persons and items, and corresponding formula? are given for the item scores. Finally, the results are applied to actual data obtained on a Moray House intelligence test, an estimate being made of the reliability of the test.

In conclusion I should like to thank Professor Godfrey H. Thomson for his help and valuable criticism in connection with this paper. I should also like to take this opportunity of thanking the Carnegie Trust for the Universities of Scotland for grants to cover the cost of the setting and printing of mathematical formula? in two papers previously published in the Society's Proceedings.

A quasi-field is denned by the postulates of a commutative algebraic field, except that the distributive law a(b + c) = ab + ac is replaced by a(b1+ … +bn)=ab1+ … +abn for a fixed integer n.

The properties of quasi-fields are investigated. The study of their ideals is reduced to the study of the ideals of a certain type of ring. A particular quasi-field is constructed formally by means of polynomial domains modulo a natural number, with addition specially defined.

Quasi-fields are connected with multiple fields—another generalisation of the conception of a commutative field, in which a fixed number of elements (> 2) co-operate symmetrically in the formation of any sum or product.

In a recent paper in these Proceedings, Dr G. C. McVittie has published some criticisms of kinematical relativity. These criticisms are to a large extent based on his formula (4.10), namely,

It must be stated at the outset that McVittie's interpretation of his derivation of (1) as a derivation of “Milne's formula for the acceleration of a ‘free particle moving in the presence of a substratum,’ for the special case of one spatial co-ordinate only” is wrong. McVittie does not derive the result, as he claims, from what he calls the “axioms of kinematical relativity” alone; he deduces it from these axioms together with an additional assumption, which is equivalent to begging the answer to the whole problem it was my object to solve. Instead of considering a free particle, as I did—that is, a particle whose motion we do not a priori know—he prescribes a priori the motion of his particle as being constrained to obey the rule, in his notation,

During the past four years I have been developing a new method of measuring the velocity of light. I had intended to defer publication until the method was perfected, but progress is becoming increasingly difficult owing to war conditions. So I give here an account of the progress already made. I am certain the method has a great future before it. There is a notice of it in a letter to Nature (Houstoun, 1938), but except for this no description of the method has yet been given.

In his recent article McVittie has criticised in most disparaging terms the analytical theory of time-keeping developed by Milne and myself in the last ten years. Milne has replied at length (see preceding paper), and it is my purpose in this note merely to touch on one or two points which he has not covered.

However, before doing so, I should like to take this opportunity of remarking that the “Kinematical Relativity,” about which McVittie has written, both in his recent article and in his monograph, “Cosmological Theory,” is not the Kinematical Relativity of Milne and myself, but is something much slighter, based, perhaps, on an incomplete understanding of the nature of the kinematical theory.

The quotient of alternants

the denominator being the difference-product of the arguments a, β, γ, …, is an important symmetric function, introduced into algebra by Jacobi in 1841.