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.

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.)

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.

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

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.

Previous theories of the recombination of ions in gases are shown to be inapplicable to high pressures. An approximate quantitative theory which takes into account all the relevant phenomena is developed. The effect of both thermal agitation and mutual attraction in determining encounters of ions of opposite sign is considered, and the frequency of encounters which lead to recombination calculated. The criterion for an encounter to lead to recombination is that the drift of the ions towards each other due to their mutual attraction must be greater than their tendency to separate due to their Brownian movement. Excellent agreement is obtained with such experimental results as are available. Preferential recombination between an ejected electron and its parent ion is also discussed.

I am indebted to various members both of the Wills Physics Laboratory and of the Cavendish Laboratory for discussion of the problems dealt with in this paper.

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

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

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.