The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the meaning of some accepted ones.

1. In den folgenden Zeilen möchte ich einige Sätze mit Beweisandeutungen mitteilen, die sich teilweise auf die Besonderheiten der Konvergenz der Partial-summen zur entwickelten Funktion teilweise auf die Eigenschaften der “Legen-dreschen Polynome” einer Potenzreihe

beziehen, in welcher die Koeffizientenfolge {αn} mehrfach monoton ist. Eine Folge {αn} heisst k-fach monoton, wenn die Zahlen der k + 1 unendlichen Folgen {αn}, {δ1αn} {δ2αn}, …,{δkαn}, n = 0, 1, 2, …, durchwegs nichtnegativ ausfallen. Hier bezeichnet δvαn die v-te Differenz von αn, d.h.

1. It is known that the distribution of the vanishing coefficients in the power series of an integral function is closely connected with the analytic properties of the function. But the following theorem has apparently not yet been noticed, although simple and elegant both in statement and in proof.

1. A “Fourier kernel” means here a function K(x) which gives rise to a formula

of the Fourier type. Thus

are Fourier kernels. If K(x) is a Fourier kernel, λ is real, and a positive, then

are Fourier kernels.

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.

The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We write

being the complex Fourier series of f (θ).

Cayley's remark that the formula by which the genus of a surface, according to Clebsch's definition, may presumably be computed leads to a negative number in the case of a cone, or a developable surface, or a ruled surface in general, has great importance in the history of the theory. But it would appear, from various indications, that, for a developable surface at least, it is more often quoted than read. I have thought therefore that the following simplifying remarks may have a use. Cayley uses formulae, due to Salmon and Cremona, without reference to the memoir where these are given in detail. Of two of these, for the number of tangents of a curve which meet it again, and for the number of triple points of the nodal curve, proofs by the theory of correspondence are extant; for the present purpose it is only necessary to have the sum of these two numbers. I do not know whether it has been remarked that there exists a remarkable formula for this sum, very similar to, and including the ordinary formula for the number of triple points of a general ruled surface (and like this probably capable of a direct proof by the theory of correspondence). For the genus of the nodal curve, deduced by Cayley from the Salmon-Cremona formulae, a proof by the theory of correspondence (in the general case, sufficient for the purpose in hand, in which i = τ = δ = δ′ = 0) is added here, which seems to have a certain interest.

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:

where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Then

provided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

Suppose that a, c and A are real, and that

Let n (X) be the number of zeros of

in ; then it is not difficult to prove that

for some K = K (A, a, c, θ). The problem in this paper is to determine the function K explicitly in terms of its variables.

The group of order 168 discovered by Klein, which it is now known can be generated by two operations E, of order 7, and ϑ, of order 2, which satisfy the relations E7 = 1, ϑ2 = 1, (Eϑ)3 = 1, (E4ϑ)4 = 1, has a vast literature. But for the most part each author pursues the matter from his own point of view; and it seems it may be useful to present a simplified approach to the theory which takes account of various possible aspects, in particular the geometrical. This is the object of the present note; for most of its contents I have found it necessary to do fresh work, so that the paper is by no means a transcript of what is already available.

It is a problem of considerable interest in the theory of surfaces to determine the irregular non-singular surface of minimum order, not referable to a scroll; in previous investigations the author has discussed the regularity or referability of surfaces in higher space, reaching the conclusion that all non-singular surfaces of order n ≤ 10 in S4 are regular or referable, with the possible exception of the surface of order n = 10 and sectional genus π = 6, which may be elliptic (pg = 0, pa = −1) or hyperelliptic (pg = 1, pa = −1). In their memoir on hyperelliptic surfaces, Enriques and Severi have obtained for the irregular hyperelliptic surface of general moduli a model 6F10 of minimum order, situated in S4, with the characters n = 10, π = 6. Using transcendental methods, Comessatti has constructed a class of irregular hyperelliptic surfaces the properties of which he has examined in detail; this class includes a member 6Π10 which is a special case of 6F10; and since Comessatti has shown that Π10 is without singularities, so also is F10, whence it follows that F10 is a solution of the proposed problem.

1. In some recent papers I have developed the theory of what I have called harmonic integrals associated with an algebraic variety. My object has been to provide an apparatus with which to investigate the properties of the classical integrals

of total differentials which are finite everywhere on an algebraic variety Vm of m dimensions

These integrals are referred to in what follows as “Abelian integrals”, as it is desirable to have a single name to denote all integrals of this type. The usual qualification “of the first kind” is omitted, and is always to be understood. In the present paper it is shown how the theory of harmonic integrals can be applied to deduce certain results concerning Abelian integrals, and a number of interesting problems are suggested by these results.

If pCε is a curve of order ε and genus p without singularities in space of three dimensions, the formula for the number of quadrisecants is well known, namely*,

Welchman has shown that the necessary reduction in this formula when the curve pCε has a point of multiplicity r with r distinct tangents, no three of which are coplanar, is

W. G. Welchman in his work on fundamental scrolls* obtains as directrix curves to such scrolls a canonical curve pK2p−2 in [p − 1] and a non-special curve pCn in [n − p]. These latter curves may not, however, be general curves regarded projectively, and it is an interesting question to find out the geometrical interpretation of their particularity. The curves C and K are, of course, in birational correspondence, and the prime sections of C correspond to the sections of K by quadrics through a contact set*, i.e. a set of points such that there is a quadric which touches K at every point of the set. For k small enough it is clear that every set of k points is a contact set and in this case the curves C are quite general. For larger k the fact that the set is a contact set simply means that, on C, the points of the bicanonical sets residual to a prime section lie themselves in primes (when k is sufficiently small the number of points in this residual set is such that they always lie in a prime).

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that

1. There are two well-known theorems on the limit of a bounded function at a point.

Montel's Theorem. Suppose that f (z) is regular for | arg z | ≤ α, | z | ≤ 1, except perhaps at z = 0, and that f(z) is bounded in that region. Suppose also that f(z) → l as z → 0 along arg z = β, where | β | < α. Then f(z) → l as z → 0 uniformly for | arg z | ≤ α − δ for every δ > 0.

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curve

which satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

In a paper which is to be published shortly formulae have been obtained for the virtual characters of the system of surfaces cut out on a primal of S4, having ordinary singularities, by adjoint primals of any order; an account of the principal results has been given elsewhere. As a first application of these formulae we propose in the present paper to investigate the canonical systems belonging to various threefolds and thence to construct the canonical models which correspond to them.

1. A set of functions {øn (x)} is said to be closed L over an interval (a, b) if for an f (x) belonging to L

implies that f(x) = 0 almost everywhere. Here f(x) is a complex valued function of the real variable x.

In the first section of this paper we illustrate the use that can be made of higher space in dealing with the problem of resolving a given Cremona transformation into the product of simpler Cremona transformations. In the second section we restrict ourselves to a particular large but finite class of Cremona transformations of [3], those of genus one, and show that these can all be built up from the four following simple types:

(1) The bilinear transformation T3,3, determined by three equations bilinear in the coordinates of the two corresponding spaces; in the most general case of this both the direct and the reverse homaloidal systems consist of cubic surfaces passing through a non-degenerate sextic of genus three;

(2) Three transformations Tn, n (n = 2, 3, 4) in which the homaloidal surfaces may in each case be obtained by taking in [4] a primal V of order n which has two (n− l)ple points, and projecting on to a given [3] from one of these points the sections of V by primes through the other; for n = 2 we have the familiar quadroquadric transformation determined by quadrics through a conic and a point.

Let F be a regular surface in ordinary space and let Z be a cut on F. Then there is on Z a point z which is the limit of a decreasing sequence of 2-dimensional Cantor manifolds lying on F.