Formulae are established below for

for both odd and even integer values of r(r>1), and x≥0; also for

this series reducing to when x is an integer

In 1920 I established the formula, a most useful one,

where γ is Euler’s constant

0·57721 56649 01532 86060 6,

and in 1920 gave a simpler proof with extension to higher powers of x(x + 1) with formulae for Σ2n+1, but not for Σ2n.

It is perhaps a rather unfortunate and possibly misleading fact that the standard list of theorems in geometry contains two which are specially labelled “Theorems on Loci” and form a section to themselves, viz. Section 3 in the I.A.A.M. schedule. It is liable to suggest that the notion of a locus and the properties of loci form an isolated subject like a chapter, say, on quadratic equations, which do not concern us until we reach it, and when this moment arrives we have merely to do our two theorems and a few examples and then may regard the subject as disposed of and consider no further reference is needed except possibly in a revision of proofs of theorems before taking School Certificate.

1. Let f(x) be a real function of a real variable x. The meanings of when λ is a positive integer, a negative integer and zero, are well known. In the first case, denotes the λth integral of f(x) with respect to x, with an arbitrary lower limit of integration. In the second case, stands for the (−λ)th differential coefficient of f(x) with respect to x. Lastly, when λ = 0, means f(x).

In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.

We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l by

where the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.

If anything were calculated to make me anxious to do justice to my theme to-night it would be the association with your society of the men to whom I owe my earliest introduction to dynamics—at St. Paul’s School to Mr Pendlebury, your Secretary, and at Trinity College, Cambridge, to Professor Forsyth, your incoming President. The interest that they implanted has survived for half a century; and the applications to sport that I propose to describe are the immediate outcome of that interest.

In his New Pathways in Science (p. 121) Sir Arthur Eddington propounds the following problem

“If A, B, C, D each speak the truth once in three times (independently), and A affirms that B denies that C declares that D is a liar, what is the probability that D is speaking the truth ?” and he has dealt with it further in the Mathematical Gazette for October 1935.

In response to Dr. Lidstone’s letter in the May number of the Gazette, I have solved the equation

z 2−(2+i)z + (4+3i)=0,

a quadratic equation with complex coefficients chosen at random, by a process, or rather a succession of processes, of complete generality.

The problem is “A affirmed that B denied that C declared that D was a liar. If A, B, C and D each speak the truth (independently) once in three times, what is the probability that D was speaking the truth ?”

This problem, first propounded, I believe, by Sir A. Eddington, has aroused considerable interest, and a solution of it was given by Sir A. Eddington in the Gazette, October 1935. His method of attack was, however, somewhat unusual, and the object of this contribution is to show how the problem may be attacked on normal lines, and a perfectly general solution obtained. For this purpose let us suppose that the chance that A speaks the truth is a, that B speaks the truth is b, and similarly with C and D.

This interesting solid has tended too much in the past to be classified as “for adults only” (and for not too many of them); but because it supplies the answer to a question which every intelligent youngster asks himself, I think it is worth while to try and smooth the path of any teacher who wants to let children of (say) 13 share in some of its treasures.