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.

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.

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.

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).

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.

Let p (x) be an essentially positive function defined in the interval 0 ≤ x ≤ π. We consider the non-linear partial differential equation

for the boundary conditions u (x, t) = 0,

for x = 0 and x = π,

It often happens that when two sets of data obtained by observation give slightly different estimates of the true value we wish to know whether the difference is significant. The usual procedure is to say that it is significant if it exceeds a certain rather arbitrary multiple of the standard error; but this is not very satisfactory, and it seems worth while to see whether any precise criterion can be obtained by a thorough application of the theory of probability.

Some theoretical results on the effect of non-normality on the t test of significance, though it is stressed that they are incomplete and not perhaps of much quantitative value, nevertheless tend to agree with the result of sampling investigations in showing that for moderate departures from normality this test may still be used with confidence, particularly for testing differences in means of equal numbers of observations.

The motion of a fluid due to the rotation of an immersed plane disc about an axis perpendicular to itself is turbulent if ωa2/ν is greater than about 105, where ω is the angular velocity of the disc, a its radius, and ν the kinematic viscosity of the fluid. The torque opposing the rotation, due to the surface friction of the fluid, was calculated originally by Kármán, both when the motion of the fluid is steady and when it is turbulent.

Recent discoveries have shown that two light quanta with total energy not less than 2mc2 can create a pair of electrons (inverse process to the ordinary annihilation). Hence it follows that ordinary black-body radiation no longer represents an equilibrium state. In thermodynamic equilibrium a certain number of electron pairs must be present, the number of which will depend upon the temperature. It can be calculated as follows.

The molecular-orbital method has been applied to a study of in its ground state and excited levels, and the relative importance of the perturbation and variational methods has been considered in some detail, as well as the effect of certain integrals which, in discussions of molecular structure, have often been neglected. It appears that the ion should exist in stable equilateral form with a nuclear distance about 0·85 Å., and that all excited levels are unstable.

Reasons are given for supposing that the molecule H3 is linear and not triangular.

The commonest form of Langmuir's adsorption isotherm is

where θ is the fraction of the surface of the solid covered by adsorbed molecules, p the gas pressure in equilibrium with the adsorbed layer and A = A (T) a function of the temperature alone. This formula is usually derived by a kinetic argument which balances the rates of deposition and re-evaporation. It is perhaps not without interest to show that formula (1) and similar formulae can be obtained directly by the usual statistical methods which evaluate all the properties of the equilibrium state of any assembly. The ordinary derivation is apt to obscure the essentially thermodynamic character of (1) and to lead one to think that its form depends on the precise mechanisms of deposition and re-evaporation, whereas in fact it depends only on the whole set of states, adsorbed and free, accessible to the molecules in question. By suitable use of the usual technique for handling assemblies obeying the Fermi-Dirac statistics the saturation effect can be naturally incorporated in the theory ab initio.

A direct method of obtaining the limitations of crystal symmetry on physical phenomena is described and discussed in greater detail for the case of diamagnetic magnetostriction. Formulae are obtained for the variation of the longitudinal and transverse effects with crystal orientation in the case of bismuth.

Recently Kohler, on the basis of some experiments by Verleger, has questioned the validity of the usual assumption that the linear Hall effect is entirely perpendicular to the current. The available experimental data are however contradictory, so a simple magneto-resistance experiment with a bismuth crystal was made, which suggested that the usual assumption was, after all, valid. The disymmetry of the Hall effect is discussed, and some of Kohler's results are generalized.

The idea of “free” electrons is purely classical, but it is convenient to introduce it also in the quantum theory of metals. In order to calculate the number of free electrons N F from a quantum mechanical model it is necessary to give an appropriate definition.

Since the pioneer work of Prof. R. H. Fowler(1) on the statistical foundations of the Debye-Hückel theory of strong electrolytes (2), several attempts have been made to obtain as much information as possible on the question of the equation of state of electrolytes from merely thermodynamical and dimensional considerations.

From experimental results on the shape of the upper portion of the continuous β ray spectra of thorium C and thorium C″ the distribution of energy is deduced. This distribution is compared with Fermi's theory wherein the shape of the curve depends on the mass of the neutrino. The comparison supports Fermi's conclusion that the mass of the neutrino is zero or certainly not more than a very small fraction of the electronic mass.

It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).

Attention has recently been directed to the measurement of the relative amplitudes of wireless echoes from the ionosphere, and some interesting methods of automatic registration have been devised (1, 2). For vertical incidence, and for wave-lengths shorter than 200 metres, visual observation shows that a single echo, even when separated from its accompanying oppositely polarized magneto-ionic component, shows considerable rapid fading (3). This fading is evident on the records published by White (1), and it is clear from these that it is not easy to determine the relative amplitudes of two echoes. This paper is an account of an apparatus which automatically integrates the amplitude of an echo over a time long compared with the period of the fading, and at the end of the period registers the integrated value on a recording galvanometer. The process is then repeated so that the final galvanometer trace represents the average value of the intensity throughout successive equal periods of time. It is further described how by using a double thread recording galvanometer, it is possible to record the averaged amplitudes of two different echoes (e.g. ordinary and extraordinary, or E and F) during alternate integrating periods.

The statistical method described in 21·4 of Fowler's Statistical Mechanics was devised to deal with assemblies in which the total numbers of elementary physical systems remained constant. The same method is applicable, with no formal changes, to assemblies in which elementary physical systems of two different sorts can be created and annihilated in pairs, like the negative and positive electrons in the theory of Dirac.