The present paper contains solutions of the tensor generalisation of Laplace's Equation. The results obtained are summarised in the two theorems enunciated in § 1. They apply only to the case when the Riemannian space forming the background of the theory is flat. In the concluding paragraph a special case is considered, and it is shown that the present theory is closely connected with Whittaker's well known general solution of the ordinary Laplace's Equation.

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equation

can be represented by integrals of the type

the path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

In Part I it has been shown that, given a contact transformation, two equations

can be derived which lead to the compatible differential equations

It will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is that

regarded as an equation in the non-commutative variables q, p which themselves satisfy the condition

We shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.

In a paper entitled “On differentiating a Matrix” H. W. Turnbull deduced some interesting and elegant results by the use of a matrix operator Ω, a matrix whose elements were the partial differential operators with respect to the elements of a square matrix x. Throughout the present paper the differential operator Ω. is used, or rather a matrix operator, which is the product of Ω and another square matrix Y.

By means of this operator in §§ 1 and 2 Bazin's matrix and Reiss's matrix are considered from the standpoint of matrices as distinct from that of determinants. Reiss's matrix is shown to be a constant times a compound of Bazin's matrix; and the latent roots of Reiss's matrix are immediately determined in terms of the latent roots of Bazin's matrix. From this result a theorem, discovered by Deruyts, is deduced as well as a more general theorem.

In a recent paper Einstein has proposed a new type of geometry in which it is possible to give an arbitrary vector in the field a displacement which conserves parallelism at a distance. It may be briefly stated as follows:

Let the metric ofthe space, supposed to be of n dimensions, be given by

At each point of the space we consider a set of n unit vectors, denoted by the n2 quantities phq, which form an orthogonal ennuple such that

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

George Alexander Gibson was born at Greenlaw, Berwickshire, in 1858. His father was a man of rare capacity and character who wrote a history of Greenlaw and taught himself Latin so as to be able to obtain the material for his history from original documents.

After attending an elementary school at Greenlaw, Gibson enrolled in 1874 as a student at the University of Glasgow, where he gained the highest places in all the classes of the Arts curriculum. At this period of his life his health was not robust and it threatened to terminate his university studies. Consequently, in 1881, he took the degree of M.A. without honours. In the following year he was able to sit the honours examination in Mathematics and Natural Philosophy, and he so distinguished himself that he was awarded the Ewing Fellowship. When the degree of D.Sc. was afterwards insti tuted and the regulation required that a candidate for it should have an honours degree in Arts or Science, his ordinary degree precluded Gibson from becoming a candidate for the doctorate which his published work would doubtless have gained for him.

By the death of Dr Pinkerton on 22nd November 1930, at the comparatively early age of 60, we lost a fellow member who in times past had given varied and valuable service to our Society, and who, as Rector of the High School of Glasgow, occupied a distinguished position among Scottish teachers.

Peter Pinkerton was born in Kilmarnock on 8th June, 1870. He received his early education in the Academy there, and was Medallist in Classics and Mathematics. In 1886 he entered Glasgow University. In 1890 he graduated M.A. with First-class Honours in Mathematics and Natural Philosophy, and was awarded the Breadalbane and John Clark Scholarships in those subjects. For two years thereafter he attended the Royal College of Science, Dublin. He took first places in Mathematics, Mechanics, Physics and Chemistry, and second place in Botany.