The expression

for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transforms

of a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and

In a recent paper, Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, as for instance when applied to the theory of the hydrogen atom or the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding, and our purpose here is to show that the practice of theoretical physicists on this point is quite consistent. The issue is a little confused because Eddington's system of mechanics is in many important respects completely different from quantum mechanics, and although Eddington's objection is to an alleged illogical practice in quantum mechanics he occasionally makes use of concepts which have no place there. Such arguments will not have any bearing on the question whether or not the practice in quantum mechanics is logically consistent—although they may have bearing on which of the two systems describes physical phenomena better.

1. I appreciate the careful reply which Dirac, Peierls and Pryce (hereafter referred to as DPP) have made to my criticism of the usual theory of the eigen-functions of a hydrogen atom. Their paper will be generally welcomed because there has not, I think, previously been available an authoritative and clear statement of what is assumed and the grounds for assuming it. Their defence, which is partly on lines I had not expected, has required very serious consideration. But the result of making the arguments more explicit is two-fold. The theory is perhaps more self-consistent than it had appeared to be; but on the other hand, the pressing need for amendment becomes too plain to be overlooked. I have endeavoured to show in the last twelve years that this amendment opens out very fertile extensions.

There are many types of problem in the study of molecular structure in which integrals involving two or more centres of force arise. We may mention the calculation of molecular energy levels, the Stark effect in molecules, the van der Waals forces between molecules, and, more recently, the internal arrangement of neutrons and protons in nuclei. During the last few years the writer has been accumulating a table of these integrals, and the explicit forms for many of them are given in this paper.

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,

and

are considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.