1. Having given the terms sn of a sequence

then Hölder's means are defined by

Cesàro's means are defined by

and a third kind, σ, of mean which will be used in this paper is defined by

1. The integral equation

where x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and that

Then, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomes

The only solutions are finite combinations of exponentials.

1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.

Logicians have long distinguished two modes of human reasoning, under the respective names of deductive and inductive reasoning. In deductive reasoning we attempt to argue from a hypothesis to its necessary consequences, which may be verifiable by observation; that is, to argue from the general to the particular. In inductive reasoning we attempt to argue from the particular, which is typically a body of observational material, to the general, which is typically a theory applicable to future experience. In statistical language we are attempting to argue from the sample to the population, from which it was drawn. Since recent statistical work has shown that this type of argument can be carried out with exactitude in a usefully large class of cases(2, 3), by means of conceptions somewhat different from those of the classical theory of probability, it may be useful briefly to restate the logical and mathematical distinctions which have to be drawn.

1. Let f(z) denote an integral function of finite order ρ. We write

It has been shown that

where hρ is a constant which depends only on ρ. We are naturally led to enquire whether some equation of the form (1.1) may be true with lim sup replaced by lim inf. In this note we show that the reverse is true. We construct an integral function of zero order for which

The proof may easily be modified to construct a function of any finite order or of infinite order for which (1.2) is satisfied.

This paper contains an attempt, begun several years ago and only partially successful, to do for the canonical curve of genus five something similar to what W. P. Milne had done for that of genus four. A fundamental feature of Milne's work was the use of a rational normal curve of order three drawn through the six points of contact of a quadric with the sextic curve; here it is not in general possible to pass a rational curve of order four through the eight points of contact. (Exceptionally it may be possible to do so and then the development follows much the same lines as in Milne's paper: only the general case is discussed in what follows.)

1. The problem of finding the number of space cubic curves which pass through p given points and have 6 – p given lines as chords has been solved by several different methods. The similar problem, in space of four dimensions, of the number of rational quartic curves which pass through p given points and have 7 – p given trisecant planes has been solved for p > 1 by F. P. White and for p = 1 by J. A. Todd. In a recent paper I discussed a similar problem for elliptic quartic curves. My present object is to apply the method of that paper to the problems mentioned above and to obtain two other sets of numbers which I believe to be new. They are (1) the number of rational normal quartic curves which have three assigned chords, pass through p assigned points, and have 3 – p assigned trisecant planes; (2) the number of curves of intersection of three quadrics in [4] which have a suitable number of assigned points and chords. The evaluation of Schubert symbols which occur in the work is done by means of a formula due to Giambelli‖. This formula is stated in a more simple form in a note at the end of this paper (7).

In a recent paper I discussed plane congruences of order two in [4] and obtained congruences of types (2, 6)1, (2, 6)2, (2, 5), (2, 4) and (2, 3). The method employed was due to Segre, who showed that a plane congruence of order two in [4] has in general a curve locus of singular points which is met by each plane in five points. Then, if we can find a curve in [4], composite or not, with an ∞2 system of quadrisecant planes of which two pass through an arbitrary point, the planes must all meet a residual curve, and we shall have obtained a congruence of the second order and a fifth incidence theorem.

1. Let

be a function regular for |z| < 1. We say that u belongs to the class Lp (p > 0) if

It has been proved by M. Riesz that, for p > 1, if u(r, θ) belongs to Lp, so does v (r, θ). Littlewood and later Hardy and Littlewood have shown that for 0 < p < 1 the theorem is no longer true: there exists an f(z) such that u(r, θ) belongs to every L1−ε and v(r, θ) belongs to no Lε(0 < ε < 1). The proof was based on the theorem (due to F. Riesz) that if, for an ε > 0, we have

then f(reiθ) exists for almost every θ.

We propose here to improve on two of the theorems given earlier, to prove some fresh theorems and to give the proof of Theorem XI stated without proof in the first part of the paper.

The linear subspaces [k] of a space [n] have been represented by Grassmann by the points of a rational locus V(n−k)(k+1), of dimension (n−k)(k+1), in ; the representation is effected by taking as independent homogeneous coordinates of a [k] in [n] the determinants of order k+1 which belong to the matrix, of k+1 rows and n+1 columns, formed by the co-ordinates of any k+1 linearly independent points of the [k].

The object of this paper is to find the characteristic functions and the characteristic numbers of the partial differential equation

valid in a domain G, and where on Γ, the boundary of G. The method employed is to transform two quadratic forms to their common self-conjugate “triangle“ of reference. The solution to the problem is given by this method in a simple manner, without the use of the integral equation theory, or the use of minimal sequences.

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfying

where A is an absolute constant.

1. Formulae obtained by enumerative methods are always liable to break down owing to the appearance of an infinity of degenerate solutions, and when this happens it is usually necessary to make a fresh start. Enumerative arguments have been used to prove the two following results for a curve of order n and genus p in space of three dimensions:

(i) The order of the ruled surface formed by trisecant lines of the curve is

(ii) The number of quadrisecant lines is

In a paper recently published I found it necessary to extend those investigations of Lord Rayleigh which deal with the instability of the surface common to two streams of fluid. The Rayleigh investigation is equivalent to a first order approximation to the flow and concludes that the initial form of the disturbance of the common surface is maintained but that its amplitude increases exponentially with the time, the form of the surface at time t being, with particular values inserted,

where a is the initial amplitude, 2π/k the wave-length, U the velocity of the upper stream, – U the velocity of the lower stream, and ρ the density of both fluids. A second order approximation to the flow for the above particular case shows that the original disturbance does not grow symmetrically, the ordinate of the common surface being given by

(A slightly amended form of this solution is given byequation III(8).) The solution suggests the initial stages in the formation of vortices. The problem was further discussed by a numerical step-by-step method and showed the formation of vortices quite clearly.

If light of a frequency which corresponds to an energy greater than the ionisation potential falls on an atom, an electron may be ejected and energy absorbed. To calculate the absorption coefficient, or the rate of absorption of energy per unit intensity of incident radiation for a given frequency, one must first choose a model for the atom. If we confine ourselves to the inner K electrons there will be two electrons in this shell for the heavier atoms, and a fairly good model of the atom is obtained by considering each electron to be moving independently in a central field of force due to the charged nucleus: i.e. we neglect electronic interaction and assume that the wave functions for the system are hydrogenic. Some writers make a partial correction for this neglect of interaction by modifying the central charge through the introduction of a screening factor which is so chosen that the minimum calculated energy required to remove one of the K electrons will agree with the experimental value provided by the K absorption edge. In general, however, the approximation is fairly good, and this is particularly so in the interior of a star where the atoms are highly ionised. It is not so good when the atom is bound as in a metal, and, of course, most of the laboratory work has been carried out on atoms in this bound state.

1. In the Lehrbuch der abzählenden Methoden Zeuthen considers the problem of determining the nature and multiplicity of the tangent lines and planes to a degenerate surface in ordinary space. The surface φ in question consists of n arbitrary planes; these meet in pairs, forming double lines, and in threes, forming triple points. The tangent cone to φ from an arbitrary point consists of plane pencils, each counted twice, as follows from the known theory of degenerate plane curves. According to Zeuthen the tangent planes are of two types:

(i) those passing through one of the triple points; there are thus such systems, and each is to be counted six times;

(ii) those passing through certain points S lying on the double lines; these depend upon the passage to the limit, it being assumed that φ is obtained as the limiting form of some nondegenerate surface. Two such points lie on each double line, so that there are n(n−1) such systems of planes.

It is well known that the planes which meet four given lines in a space of four dimensions meet a fifth line, determined by the first four. Also the trisecant planes of a rational quartic curve in [4] which meet a line meet another rational quartic curve. These two theorems are of the same type and may conveniently be called “fifth incidence theorems”.

An approximate calculation of the electromagnetic field of a vertical dipole at the surface of a conducting earth, for small angles of the radius vector with the horizontal, was given by Sommerfield; the case of large angles with the horizontal has been studied by a number of writers. It is proposed here to develop formulae for the vertical dipole by a method which takes into account the singularities of the integrand of a certain integral more accurately than is done by Sommerfield; the analysis is developed especially for small horizontal angles and small numerical distances.