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.

If an atom is excited by electrons incident upon it, the wave functions for the outgoing electrons must be calculated in the field of the excited atom. Such calculations have been made for helium excited to the singlet 2p state, and it is found that the outgoing waves differ more from plane waves than the incident waves.

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.

The electron theory of metals, established in its present form mainly by Pauli, Sommerfeld and Bloch, makes it possible to classify the phenomena connected with metals into two main divisions, which may be called first order effects and second order effects, according to whether the effect depends on the first or second approximation to the energy of a degenerate electron gas at a given temperature. Examples of the first group are the constant paramagnetism of the alkalis and Volta contact potentials. Examples of the second group are the specific heat of the electrons and all thermoelectric effects. The temperature dependence of the electrical conductivity should be included in the group of first order effects, since here the temperature is introduced through the lattice wave motion. The limits of the existing theory are now easily described. First order effects are accounted for with success, second order effects are in the main far from being adequately covered by the theory. To see clearly the reason for this, one must examine briefly the basis of the existing theory. In the first approximation the metal is regarded as composed of two independent systems; a system of lattice vibrations, and a system of electrons free to move in a given space periodic field of potential. In this way by dealing with the two systems quite independently one can reproduce many properties of metals, for example, Debye's theory of specific heats. The theory of pure lattice heat conductivity is concerned only with the former system, while the constant paramagnetism of the alkalis, and the fact that the electrons contribute only a very small amount to the total specific heat of the metal can be accounted for by considering the latter system only.

This paper describes a repetition of J. B. Johnson's determination of Boltzmann's constant by observing the thermal agitation of electrons in a conductor. The principle of the measurement is described and the various sources of error are examined separately. The attenuator system of the calibrating circuit is discussed in detail and mutual agreement established between the three different attenuating systems. The valve amplifier had a maximum response which could be arranged to occur at any frequency between 1000 and 4000 cycles/sec. The mean value of 52 determinations of Boltzmann's constant (with the amplifier set to respond to one of four frequencies within the range 1 to 4 k.c./s.) was found to be 1·361 × 10−16 ergs per degree, with a probable error of 1·42%: this value is 0·8% less than the accepted value of 1·372 × 10−16 ergs per degree. The investigation developed out of a research initiated with a directly practical objective; we desire to express our thanks to the Hebdomadal Council of the University of Oxford, who by a grant to one of us have enabled this section of the work to be completed.

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.

The solution of linear differential equations by the method of Frobenius is a straightforward matter consisting merely in the substitution of a power series and obtaining the coefficients from a recurrence relation. The corresponding method when applied to difference equations is laborious and leads in general to divergent series from which convergent solutions can be obtained by a method due to Birkhoff(1). Actually a more appropriate method is to find a factorial series, but even here the direct substitution of such a series requires repeated transformations and throws little light on the structure of the equation. A method of symbolic operators was originated by Boole(2), but owing to the restricted definition of the operators it has a very limited scope. In the present paper a more general interpretation is given to the operators, and their application to a certain class of linear difference equations with rational coefficients is discussed.

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.

The present paper is devoted to a consideration of a type of surface which appeared incidentally in some recent work and which seems worth separate notice for its own intrinsic interest. The surface in question, which lies in space of four dimensions, is one that possesses a triple curve containing pinch-points and quadruple points, and as such is a particular case of a general surface in [4] with the same projective characters. The results given here are illustrative of those contained in the papers referred to, and with few exceptions the main facts which are used in the present discussion are quoted from the same source. In the section dealing with rational surfaces some novel results are obtained in connection with the degeneration of a plane curve into two portions, one of which is counted twice.

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.

1. Many experiments have shown that the natural heterogeneous β-rays from any radioactive source are absorbed according to a law which is exponential for small thicknesses of absorber but which falls below the exponential for larger thicknesses, giving, finally, a quite definite ‘kink’. The positions of these kinks have been measured by the author(1), by Feather (2) and by others. Experiments on the continuous spectra of β-rays (e.g. Gurney(3)) seem to indicate a definite high-velocity limit to the continuous spectrum; and experiments on the absorption of homogeneous β-rays (e.g. Madgwick(4)) show a definite extrapolated range for the β-rays. So one might expect the ‘kinks’ found to represent the extrapolated ranges of the β-rays of highest velocity. The author (1), using the extrapolated ranges of Madgwick(4) and Varder(5), finds this interpretation of the kinks to give values of the end-points agreeing with the direct experimental values of Gurney(6). Feather(2), on the other hand, uses the results of the kinks in certain cases to give an empirical relationship between the kinks and the end-points. Feather's relationship is purely empirical and is independent of any interpretation given to the kinks; for the purpose of using the kinks to find the end-points, Feather's relationship is sufficient, but the actual interpretation of the kinks is of interest. Feather(7) has considered the matter from the theoretical standpoint, and he finds no reason to expect any abrupt discontinuity; he interprets the apparent kink as the limit of the measurable β-ray effect.

Equations (3) and (4) describe the changes undergone by a Mendelian population mating at random, and under intense selection.

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

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. Among the equations arising in the theory of natural selection is the finite difference equation

where k is a constant which may have any value between 1 and − ∞ inclusive, but is often small. Its solution is discussed in the accompanying paper(1). It is a particular case of the equation

where Φ(x) is a known one-valued function of x. When k is small this may obviously be solved approximately by treating it as a differential equation

whence

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.

The marked disagreement between the specific heats at high temperatures of the diatomic gases as found by sound velocity measurements, and those calculated from theoretical considerations, has for long been a source of perplexity to statistical mechanists; a perplexity all the greater in view of the simple nature of the theoretical calculations which appeared to form as direct a test of Boltzmann's hypothesis and the theory of band spectra as was possible to large scale measurements. One way out of the difficulty is to attribute the differences to the experimental difficulties inherent in the sound velocity method of measurement at high temperatures(1), but it cannot be denied that there is a certain measure of agreement between the separate velocity determinations. It consequently becomes necessary to examine the theory.