1. Let

be a numerical series. If for sufficiently small h > 0 the series

is convergent, we can form the upper and lower limits of J (h) as h → 0. These limits are called respectively the upper and lower sums (R, 1) of the series (1). For the purposes of the present paper it will be convenient to consider a more extended definition of these upper and lower sums. We shall suppose that for sufficiently small h the series J (h) is summable by Poisson's method. We denote the Poisson sum by PJ (h). The upper and lower limits of PJ (h) as h → 0 will be called the upper and lower sums (R, 1) of the series (1).

The theory of the intersection-complex Ch · Ck, due originally to Veblen, Weyl, Alexander and Lefschetz, has recently been extended by Flexner to cover intersections on a “topological n-manifold” (locally Cartesian n-space) in the case h + k = n. In his theory the whole of Lefschetz's work for ordinary simplicial manifolds is presupposed, including the rather difficult approximations. The object of the present paper is to provide a combinatory theory of the general intersection, (h + k ≥ n), which would serve equally well as a basis for Flexner's work, and avoids the wasteful process of using two independent approximations.

Engineering is not, as some are mistakenly apt to imagine, a comparatively recent introduction into Cambridge curricula. Professor William Farish, B.D. of Magdalene, Jacksonian Professor and our first President, contributed the first paper in Vol. I of our Transactions, “On Isometrical Perspective,” a method of which he appears to be the inventor. The reason for the invention he explains. He was evidently teaching engineering—or applied mechanics—so he wanted numerous models of machines and “would have found it difficult to procure a warehouse large enough to contain them.” So he obtained a number of interchangeable parts: loose brass wheels all gearing together, pulleys, axles, bars and frames from which different models could be built up, in fact he invented the forerunner of Meccano and all its competitors. His assistants needed drawings showing them how to assemble the parts for these different models, and he devised this very simple and readily comprehensible method for making such drawings.

The principal object of the present paper is to give a set of sufficient conditions, of considerable generality, under which a plane curve assigned to pass through certain points with given multiplicities shall be irreducible. The case considered is that in which the assigned multiple points of the curve are of such generality that the conditions presented to the curve at these points are independent. We shall shew that subject to certain explicit restrictions on the points, which ensure, for example, that no three points shall be collinear whose prescribed multiplicities in aggregate exceed the order of the curve, the curve determined by the points will be irreducible if its freedom is not negative, save in the case when it degenerates into a repeated elliptic curve. A familiar example of this last case arises in the problem of constructing a sextic with nine assigned nodes; in general the curve in question consists of the unique cubic through the nine points, counted twice.

1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.

In view of the interest now taken in Cremona Transformations in more than three dimensions, some account seems desirable of such transformations in four dimensions. Many of the ideas in two and three dimensions are capable of immediate extension. In this paper attention is directed to the new possibilities in regard to the Principal and Fundamental Systems. The paper is written on the lines and with the notation of Miss Hudson's book.

In an investigation which I hope to publish shortly, I think I have been able to improve my theory of the constant hc/2πe2 and to bring it at last into a precise form. No alteration is made in the value 137 obtained in the work already published. The recent advance has been mainly due to the fresh light thrown on the foundations of wave-mechanics by Dr Dirac's book. With a fuller understanding of the “theory of 137” it has been possible to discern opportunities for extension in several directions, and it is with these developments that the present paper deals. They are still in a rudimentary state; but since the theory appears to give correctly either accurate or approximate values of the masses of the electron, the helium atom, and the cosmos in terms of the mass of the proton, it would seem to be on the right lines. Moreover the principle of “ignoration of degrees of freedom” on which the numerical predictions depend is strongly suggested by the theory of the constant 137. If my view is right the only arbitrary constant of nature is the number of particles in the universe—if the number is arbitrary.

1. The analogue for trigonometric integrals of de la Vallée Poussin's classical theorem concerning the uniqueness of the trigonometrical development has recently been obtained by M. Jacob in the following form.

The considerations set forth in the present paper are valid for the most part in parallel form for integration in space of any number of dimensions, and in the widest measure for any modification of the process that may be devised naturally: integration of functions not necessarily one-valued, with respect to functions of sets, or functions of ∑, not necessarily unchanged by subdivision, in fields as wide or as restricted as may be convenient, as long as we maintain the possibility of forming subdivisions of arbitrarily small “norm,” and of splitting up such a subdivision arbitrarily into two parts, each of which may be subdivided independently in any manner, to give a subdivision of the whole. These conditions are essential for the proper working of the theory, and in particular for the validity in more dimensions of Hyslop's method referred to in 5.

1. The problem which I propose to solve is that of finding the number of quartic curves of intersection of two quadrics which pass through p points and have q lines as chords, where p + q = 8. There are ∞16 elliptic quartics in space; to contain a line as a chord is two conditions, and to pass through a point is two conditions, so we should expect a finite number of solutions. Throughout this paper I shall refer to an elliptic quartic curve in space of three dimensions simply as a “quartic.” I shall denote the number of solutions for a particular value of p by np.

1. It is possible to treat the excitation of an atom by an α-particle in two ways; we may either solve the Schrödinger equation for the system consisting of the α-particle and the atom, or we may, on account of the great mass of the α-particle, treat it as a moving centre of force, and solve the Schrödinger equation for the electrons in the field of the α-particle and nucleus, If the α-particle has velocity υ greater than the orbital velocities of the electrons, it is possible to obtain approximate formulae for the excitation probabilities by both methods; in the former case by the well known Born method, and in the latter case by a method first used in this connection by Gaunt*, and which is essentially the same as the method of variation of constants. The two methods give formally very different formulae for the excitation probabilities; it is the purpose of this paper to show that they are in fact identical if the ratio of the mass of the electron to that of the α-particle be considered vanishingly small.

1. In a recent paper I have given some definite integrals involving Legendre functions which, as a limiting case, give known results involving Bessel functions. In another paper I have shown how some integrals involving Bessel functions can be obtained from Bateman's integral

and the well-known expansion

The principles and general properties indicated in the first part of the present theory are those which belong most strictly to the province of the “algebra of many-valued quantities,” by their practically exclusive reliance on notions of equality, union, and passage to the limit in this domain.

The word “freedom” is used in this paper with the same meaning as “Konstantenzahl"—the number of free constants—and corresponds very roughly to the term “degrees of freedom” as used in elementary mechanics. “Freedom of a manifold” is used as a convenient abbreviation for “freedom of the class of manifolds of the same specification as the given one”. At the outset the idea is rather vague; in the background there is an intuitive notion of a definite number associated with any given manifold—the number of conditions that manifolds of the same specification can be made to satisfy. This paper is intented to give some precision to this idea, and to formulate rules which will enable the freedom of a given manifold to be calculated.

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.

1. Mordell has recently proved that if A, B, C, D, P, Q are any real constants such that Δ = AD − BC ≠ 0, and if ΔBC ≠ 0, then the inequalities

are satisfied by integer values of x and y.

In a recent paper Mr W. G. Welchman has devised an extremely simple method for determining the number of elliptic quartic curves in space which pass through p given points and have 8–p assigned lines as chords. In the present note this method will be applied, with certain necessary extensions, to some cases in which the curves are subject to the condition of meeting certain lines in one point only, in addition to conditions of the previous type. The method in principle is perfectly straightforward, but circumstances arise which make its application somewhat limited.

Some applications of Fourier series in the analytic theory of numbers.

Page 589, equation (3–10), after “k>0” insert “and 0<R(s)< 1,” and for “2nπi/k” read “2nπix/k.”

Page 589, equation (3·11), for

Add also ‘The evaluation of the integrals given in (3·11) is obvious when 0 < R (s) < 1, and then holds also for 0 < R (s) < 2 by the theory of analytic continuation.”

1. In a paper published recently in these Proceedings, I have proved the formulae

where R(m)>0;

where R(m) > −½; and

where R(m) >− 1. In each case n is unrestricted.

In this paper it is shown how all the regular polytopes (including the Kepler-Poinsot polyhedra and Hess's analogous star polytopes in four dimensions) arise as rational solutions of trigonometrical equations.