The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

In spite of a large amount of general theory, it remains a surprising fact that the number of known irregular surfaces in space of three dimensions is quite small. Moreover, although the evaluation of the arithmetical genus pα is comparatively direct, it is often a matter of considerable difficulty to calculate the geometrical genus pg. This paper deals with a number of surfaces for which the equations of the canonical system can be written down and their number counted. An extension to primals in higher space is also suggested for the simplest case.

James Clerk Maxwell, in his days of early development, made a practice of communicating his progress in ideas by informal letters to his scientific friends G. G. Stokes and W. Thomson, who were in the habit of preserving their correspondence. The record, so far as revealed in the letters to Stokes, has been published in volume 2 of Prof. Stokes' Scientific Correspondence. The letters which are here printed have emerged among Lord Kelvin's manuscript remains. They had been arranged apparently by Prof. S. P. Thompson when he was preparing his biography of Lord Kelvin's practical activities. I find that they had been examined by myself when a project of publishing Lord Kelvin's scientific correspondence was contemplated, after the manner of that of Stokes; which afterwards proved to be impracticable, as the material had largely been skimmed over.

In his chapter on correspondences between algebraic curves Prof. Baker has raised a problem concerning the possibility, when we are given the equations of Hurwitz for a correspondence between two algebraic curves, of obtaining therefrom a reduction of the everywhere finite integrals on either curve into complementary regular defective systems of integrals. The problem is stated as an unproved theorem, an exact formulation of which is given below. The object of the present note is to give a proof of this theorem on the lines of Prof. Baker's chapter.

There is a theorem for three circles in a plane, that if three tangents, each of two of these circles, either all transverse or one transverse and two direct, can be drawn to meet in a point, then the three tangents, each of two of the circles, respectively conjugate to those first taken, likewise meet in a point. The theorem was stated by Quidde, with a proof for the necessity of the condition as to the tangents to be taken, in a paper designed to establish Steiner's solution of Malfatti's problem. Casey gives the theorem with omission of the condition for the character of the tangents, as does Salmon, who, however, gives a proof depending on the right choice of certain square roots which enter. Quidde's theorem is stated, accurately, in the Nouvelles Annales, and a simple metrical proof, from the diagram drawn (essentially Quidde's, see 6 below) is given later in the same Journal by Mannheim; this is practically repeated by Hart. Recently, Prof. Neville, emphasizing the necessity of the condition for the character of the tangents, has called attention to Quidde's paper.

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.

According to Carnot's postulate heat can give rise to work only by falling to lower temperature. This negation ensures that at each temperature a definite physical system possesses a work-function A, named by W. Thomson its available, energy at that temperature. This aspect of Carnot's postulate was enforced especially in Thomson and Tait's Nat. Phil. (1867). These available energies at different temperatures combine into one more general function, the isothermal available energy, which is a function of temperature θ as well as configuration. (Cf. the Minkowski condensation of personal spaces and times into a single universal space-time.) Thus we are entitled to assert the equation

where Ψ1r is the force exerted by the coordinate ψr of configuration. The final term in δA though regular has not to do with ostensible work: and −η, as yet arbitrary, here equal to A/θ, may be described as the thermal capacity or specific heat of available energy.

Let a1, a2, … be a sequence of variables. Write

We use the notation Bn = Bn (a1, …, an) to indicate the variables on which Bn depends.

The groups of rotations that transform the regular polygons and polyhedra into themselves have. been studied for many years. Lately, increasing interest has been shown in the “extended” groups, which include reflections (and other congruent transformations of negative determinant). Todd has proved that every such group can be defined abstractly in the form

This group is denoted by [k1, k2, …, kn−1], and is the complete (extended) group of symmetries of either of the reciprocal n.-dimensional polytopes {k1, k2,…, kn−1}, {kn−1, kn−2,…, k1}. There is a sense in which these statements hold for arbitrarily large values of the k's. But here we are concerned only with the cases where the groups and the polytopes are finite. The finite groups are

[k] is simply isomorphic with the dihedral group of order 2k (e.g. [2], the Vierergruppe). [3, 3,…, 3] with n − 1 threes, or briefly [3n−1], is simply isomorphic with the symmetric group of order (n + 1)!.

In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:

(1) Regular, non-rational.

(2) Rational.

(3) Irregular, non-referable.

(4) Referable.

1. In a former paper we solved the following problem (Problem I). Given k + 1 real numbers co, …, ck, to find necessary and sufficient conditions that there shall exist a function f(x) in (0, 1) which satisfies the conditions

Let d(m) denote the number of divisors of the integer m. Chowla has conjectured that the integers for which d(m + 1) > d(m) have density ½. In this paper I prove and generalize this conjecture. I prove in § 1 a corresponding result for a general class of functions f(m), and in § 2 the result for d(m) which is not included among the f(m). I employ the method used in my paper: “On the density of some sequences of numbers.”

On a surface F′ an algebraic self-correspondence T of period n defines a cyclic involution In of sets of n points. Then if there exists a surface F whose points are in (1, 1) correspondence with the sets of In, the surfaces F, F′ will be said to be in (1, n) cyclic correspondence. The purpose of the present paper is to show that, when n is a prime number and with certain restrictions upon the united curve of the self-correspondence T, the irregularities of the surfaces F and F′ are equal.

The Abel sum of the series can be written in the form

where

This suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the form

where

A1(u) may conveniently be described as the “Abel transform” of A(u).

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation

with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

1. In 1891, Castelnuovo suggested that there may exist canonical surfaces which consist of three sheets covering the same surface. In particular, he remarked that, if the canonical curves of a surface contain a then the canonical model of the surface is certainly a three-sheeted surface. An example is the quintic surface with a tacnode, in space of three dimensions; its canonical model is a triple plane, branching along a curve of order 12 with twenty-four cusps which lie on a quartic.

1. Given a sequence of variables a1, a2, …, we define a sequence of polynomials B1, B2, … by

For every positive integer n, Bn is a polynomial in a1, …, an, with rational coefficients. We write Bn ≡ Bn (a1, …, an) to indicate the variables on which Bn depends. From the polynomials Bn, we form a new sequence of polynomials Dn in accordance with the following definitions.