En prenant connaissance d'un mémoire récent de M. G. Doetsch(2), mon attention a été ramenée sur quelques remarques de nature élémentaire que la lecture de la note de M. Hardy, citée ci-dessus, m'avait suggérées et qui me paraissent apporter quelque clarté sur l'origine des résultats obtenus par ces auteurs relativement à la composition des noyaux de Fourier. Des remarques analogues sont contenues dans un article de M. Hermann Kober qui paraitra prochainement(3) et dont M. Hardy m'a obligeamment communiqué les épreuves. La voie que j'ai suivie diffère de celle de M. Kober; je l'expose brièvement ci-dessous.

This paper is concerned with the extension to [2n] of a well-known symmetrical configuration in [4], namely, that of three rational quartic curves, with six points in common, having the property that a trisecant plane of any one which meets a second also meets the third.

The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.

The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of order r, in space [r], say the curve cr [r], there be an involution ∞k, of sets of m points, expressed, suppose, by an equation

then the spaces [r − 1], formed from r points of any one of the polyhedra of m points, are an aggregate ∞k of primes of this space [r], which is of class (m − k, m − r), the notation (p, q) meaning the binomial coefficient p! / q!(p − q)!. By Meyer, the conditions k < r ≤ m are supposed to be satisfied. But there is a theorem for r < k ≤ m − 1, relating to selected [r − 1], formed from r points of any one of the polyhedra of m points. The general theorem may be formulated thus: In a space [r], the equation of any prime may be expressed by λu + μv +... + ρw = 0, where u = 0, v = 0, …w = 0 are any r + 1 given independent primes, and λ, μ, …ρ are coefficients which may be described as prime coordinates of the [r].

It is a familiar fact that the arithmetic genus pa and the arithmetic linear genus ω of a general surface are linear functions of its four projective characters; and we find by direct calculation that a similar property holds for the numerical invariants of a general threefold. The question thus arises, whether this result can be established a priori for any algebraic variety Vk of general type, since in that case we should have a simple means of determining its numerical invariants. It has been shown by Severi that, subject to a certain assumption, the arithmetic genus pk of Vk is a function of its projective characters, while it is known that, for k ≤ 4, pk coincides with the arithmetic genus Pa obtained by the second definition (§ 5). In the present paper we obtain, by using Severi's postulate, expressions for the arithmetic genera of a V3 and a V4 in terms of their projective characters. We obtain also the characters of their virtual canonical systems and hence derive formulae for the relative invariants Ωi. For this purpose we replace certain projective characters of Vk by others which are more easily computed and better adapted to a simple notation.

1. Prof. Fekete, in a letter to the author, wrote: “…Satz von Myrberg über den Zusammenhang zwischen dem transfiniten Durchmesser und dem logarithmischen Mass zugehörig zum Exponenten 1 + ε, ε > 0 (vgl. Satz 10 in seiner Acta Arbeit).…Ich würde einen elementaren Beweis für den letzteren Satz sehr begrüssen.” The object of this note is to supply such a proof but, since the concepts involved may not all be matters of common knowledge, I shall begin by outlining the necessary definitions, the relevant known results, the nature of the theorem and its relation to what is known.

The famous theorem of Schnirelmann states that a constant c exists such that every integer greater than one may be expressed as the sum of at most c primes. Recently, Heilbronn, Landau and Scherk proved that this holds with c = 71. Probably the true value of c is 3. Another well-known theorem (Hilbert's solution of Waring's problem) is that every integer is the sum of a bounded number of positive kth powers. If we omit the restriction that all the kth powers are positive, the problem is referred to as the easier Waring problem and the proof of the result is then much simpler.

1. In the present note we obtain the postulation of a multiple surface of S4 for primals of sufficiently high order; the method is that previously used by Campedelli to determine the postulation of a multiple curve of S3.

The object of this paper is to make more precise the results given by Enriques-Campedelli for canonical surfaces, that is, surfaces whose prime sections are the canonical curves. We consider the case of regular surfaces in space of three dimensions (pg = pn = 4), and of regular multiple planes (pg = pn = 3).

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

In the first part of this paper I announced some new Pflaster theorems for arbitrary r-dimensional closed sets lying in the Euclidean space Rn. In § III I proved them for the special case of an r-dimensional closed set F linked (rel a neighbourhood U) with an (n – r – 1)-dimensional spherical cycle. I shall now prove these theorems in the general case of a quite arbitrary closed set F.

1. While the general theory of surfaces has received a great deal of attention in recent years, there remain a number of difficulties in the application of the results of that theory to the study of particular surfaces. I propose to discuss here certain details which, from their nature, do not seem to be amenable to very general treatment. I shall therefore consider a number of examples with a view to illustrating the way in which certain classes of surfaces may be expected to behave.

This note is concerned with the relations between the invariants Ωi (i = 0,1,…, d – 1) of the canonical system on an algebraic Vd. Here Ω0 is the grade of the canonical system, and Ωi is the virtual arithmetic genus of the Vi of intersection of d – i canonical Vd–1's.

The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.

Let a1, a2, …, an be a set of n positive integers. Then it is easily seen that the set of positive integers not divisible by any aν has a density, i.e. that if Nn(z) is the number of such integers not exceeding z, then z−1Nn(z) tends to a limit when z → ∞; and that

where

and where [u1, …, uμ] denotes the least positive common multiple of the positive integers u1, …, uμ.

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1, k2, …, kn−1] is denoted by [k1, k2, …, kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1, k2, …, kn−1}, {kn−1, kn−2, …, k1}; e.g., [3, 4]′ is the octahedral group.

1. It often happens that we have a series of observed data for different values of the argument and with known standard errors, and wish to remove the random errors as far as possible before interpolation. In many cases previous considerations suggest a form for the true value of the function; then the best method is to determine the adjustable parameters in this function by least squares. If the number required is not initially known, as for a polynomial where we do not know how many terms to retain, the number can be determined by finding out at what stage the introduction of a new parameter is not supported by the observations*. In many other cases, again, existing theory does not suggest a form for the solution, but the observations themselves suggest one when the departures from some simple function are found to be much less than the whole range of variation and to be consistent with the standard errors. The same method can then be used. There are, however, further cases where no simple function is suggested either by previous theory or by the data themselves. Even in these the presence of errors in the data is expected. If ε is the actual error of any observed value and σ the standard error, the expectation of Σε2/σ2 is equal to the number of observed values. Part, at least, of any irregularity in the data, such as is revealed by the divided differences, can therefore be attributed to random error, and we are entitled to try to reduce it.

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:

Lemma 1.

provided that R(m) > 0.