James Gregory was the third son of the Rev. John Gregory, minister of Drumoak, a small parish near Aberdeen. His mother was the daughter of David Anderson of Finzeach in Aberdeenshire, and related to Alexander Anderson, a friend of Vieta and a teacher of mathematics in Paris. Gregory is said to have received his first lessons in mathematics from his mother, but in due course he passed on first to the Grammar School and then to Marischal College, Aberdeen, where he graduated. In 1663 his Optica Promota was published in London, and he spent some time in that city after the publication of his book in the hope of securing facilities for constructing a telescope on the principles he had laid down in the Optica. His efforts were however unsuccessful, and he went to Italy where he continued his mathematical studies. After a residence of three years in Padua he returned to Scotland in 1668. In 1669 he was appointed to the Chair of Mathematics at St Andrews; in that position he had a busy and, as the years passed, a rather troubled life, so that he was glad to accept a call in 1674 to be Professor of Mathematics at Edinburgh where, as he says in a letter to a friend in Paris, “my salary is double and my encouragements much greater.” His Edinburgh professorship was however very brief as he died in October 1675. An interesting sketch of his life is given by Agnes Grainger Stewart in The Academic Gregories, a volume of the “Famous Scots Series.”

The following paper is a continuation of one read before the Society some years ago, and published in the Proceedings, Vol. XXXIV. (Part 2), Session 1915–1916. The results of that paper, more especially those summarised in Art. 14, and those of Arts. 17, 18 will be assumed.

In the answer to the book-work question, set in a recent examination to investigate the volume of a pyramid, one candidate stated that the three tetrahedra into which a triangular prism can be divided are congruent, instead of only equal in volume. It was an interesting question to determine the conditions in order that the three tetrahedra should be congruent, and this led to the wider problem – to determine what tetrahedra can fill up space by repetitions. An exhaustive examination of this required one to keep an open mind as regards whether space is euclidean, elliptic, or hyperbolic, and then to pick out the forms which exist in euclidean space.

In two-dimensional discontinuous fluid motion one point of considerable importance has not hitherto been given sufficient attention. I raise it formally in a paper to be published soon (Fluid Motion past Circular Barriers, Scripta Universitatis atque Bibliothecae Heirosolymitanarum, 1923, Vol. I., XI., 1–14) in the following manner. Given the form of the barrier by means of, say, the radius of curvature in terms of the angle of contingence, how does the solution take into account the angular extent of the barrier? Clearly barriers which are defined by the same curve, but differ in the extent of curve used, must necessarily give rise to different solutions. Further, there must be a limiting extent of barrier, so that if it extends beyond this limit the part of the barrier in excess must lie in the “dead” fluid.

Suppose that as a result of observation or experience of some kind we have obtained a set of values of a variable u corresponding to equidistant values of its argument; let these be denoted by u1, u2, … un If they have been derived from observations of some natural phenomenon, they will be affected by errors of observation; if they are statistical data derived from the examination of a comparatively small field, they will be affected by irregularities arising from the accidental peculiarities of the field; that is to say, if we examine another field and derive a set of values of u from it, the sets of values of u derived from the two fields will not in general agree with each other In any case, if we form a table of the differences δu1 = u2 – u1, δu2 = u3 – u2, …, δ2u1 = δu2 − δu1, etc., it will generally be found that these differences are so irregular that the difference-table cannot be used for the purposes to which a difference-table is usually put, viz., finding interpolated values of u, or differential coefficients of u with respect to its argument, or definite integrals involving u; before we can use the difference-tables we must perform a process of “smoothing,” that is to say, we must find another sequence u1′, u2′, u3′, …, un′, whose terms differ as little as possible from the terms of the sequence u1, u2, … un, but which has regular differences. This smoothing process, leading to the formation of u1′, u2′ … un′, is called the graduation or adjustment of the observations.

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equation

where α, β, γ are functions of x and y.

Associated Legendre Functions as Integrals involving Bessel Functions. Let

,

where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

The periodic solutions of the linear differential equation

,

which reduce to Mathieu functions when v = 0 or 1, will be known as the associated Mathieu functions. The significance of this terminology will appear in the following section.

Professor Whittaker, in a paper entitled “On Tubes of Electromagnetic Force” {see Proceedings of the Royal Society of Edinburgh, Vol. XLII., Part I. (No 1)}, introduces certain surfaces, which he names calamoids, in connection with an electromagnetic field in the four-dimensional world of space-time. The calamoids consist of “a convariant family of surfaces which when the field is purely electrostatic or purely magnetostatic reduce to the ordinary Faraday tubes of force.” Professor Whittaker, in the paper referred to, also introduces two sets of surfaces, each a covariant family of ∞2 surfaces, one of them named the electropotential surfaces, and the other family the magnetopotential surfaces of the electromagnetic field. The electropotential surfaces and the magnetopotential surfaces are shown to be everywhere absolutely orthogonal. (One member of each family meeting at a point, any line from this point in the one family is orthogonal to every line through the point in the other family). Moreover, a “calamoid, at every one of its points, is half-parallel and half-orthogonal to the electropotential surface which passes through the point, and is also half-parallel and half-orthogonal to the magnetopotential surface which passes through the point.”

The point known in elementary geometry by the name of Tarry was first discussed by that writer as the point of concurrence of the perpendiculars respectively from the vertices of the base-triangle to the corresponding sides of Brocards first triangle. Tarry's point is the point of the circumcircle diametrically opposite to Steiner's point, which is the fourth common point of the circumcircle and Steiner's circumellipse

Gordan's Theorem, that the complete system of irreducible concomitants of a given form is finite, has been extended by Hilbert to cover wide ranges of systems of variables Gordan and Study have dealt shortly with the problem for double binary forms, approaching the subject through the theory of binary types. The following pages give a proof after the manner of Gordan's proof for ordinary binary forms, which has the advantage of providing a practical method for constructing the complete system. As illustrations, the cases of the (1, 2), (2, 2), and (3, 3) forms are considered.

In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of s and g for which a real solution exists, except for those values for which s = 2g and k = 1, but that, on the other hand, the series solution is convergent and represents the motion only so long as

for values of s and g for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of s and g which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.

It has been shown in the previous three parts of this work, that the whole question of the convergence of the series solution, for the particular dynamical system under consideration, has turned upon the cubic equation

Before proceeding to generalize the results it will be shown how this cubic equation may be derived in a slightly different fashion.