Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound of

for integral values of u, v, w, not all zero. I proved a few years ago (1) that

more precisely, that

except when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to

(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

In a recent note I attempted to obtain the postulation formula for the Grassmannian of k-spaces in [n] by the consideration of forms of a certain type in k + 1 sets of r + 1 homogeneous variables, which I called k-connexes. My attempt was not entirely successful; I obtained a formula for k-connexes which suggested what the required postulation formula should be, but was unable to prove it. D. E. Littlewood has now written a paper to show that my problem is intimately connected with the theory of invariant matrices, and has thereby established the truth of the postulation formula which I had conjectured. Littlewood's proof requires a considerable knowledge of the theory of invariant matrices, and this paper results from an attempt to re-write his proof in a form which is intelligible to a student not having this specialized knowledge. Prof. H. W. Turnbull has pointed out to me the importance of the so-called k-connexes in the theory of forms, particularly in connexion with the Gordan-Capelli series, and for this reason I am taking the k-connexes as the principal topic of this paper, leaving the deduction of certain postulation formulae which are the more immediate concern of a geometer to the end.

On an algebraic surface f of order n in space of three dimensions, the canonical system | k | of curves is traced by all those surfaces π of order n − 4 which fulfil certain conditions of adjunction at the singularities of f: for example, which pass simply through the double curves and through the isolated tacnodes of f; which pass doubly through the isolated fourfold points of f; and so on. The assigned fixed points and curves of the adjoint surfaces ø at these singularities of f are not taken to be part of the canonical system | k |; but | k | may have unassigned fixed parts e. Three cases are usually (3) distinguished.

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.

In the construction of the seismological tables it has been found that the times of several pulses at a given distance show a well-marked maximum frequency for each, about which the bulk of the observations are spread with a standard deviation of 1·5−3 sec. For others there was only a vague concentration, spread over about half a minute (Jeffreys and Bullen, 1936–40). The theoretical times of the latter type could all be calculated from those of the former by using the principle of stationary time for small variations of the path, and the observed times might be either shorter or longer than the theoretical ones. There was a complete association between the type of distribution of the residuals and the nature of the path. Whenever the time as calculated was a minimum for all small variations of the path, the distribution of residuals was of the former type; when it was stationary without being either a true minimum or a maximum it was of the latter.

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,

Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

Assuming that the solute and solvent molecules in a solution can be regarded as occupying sites on a regular lattice and that the potential energy arises from interactions between molecules which occupy closest neighbour sites, the vapour-pressure equations have been determined for solutions in which each solute molecule contains three groups (or submolecules) and occupies three lattice sites in such a way that successive submolecules occupy closest neighbour sites on the lattice while each solvent molecule occupies only one site. The vapour-pressure equations are compared with those which have already been obtained by Fowler and Guggenheim for the case in which each solute molecule contains two submolecules in order to determine the effect on the form of the vapour-pressure equations of the number of submolecules in each solute molecule. This enables the determination of the vapour-pressure equations when each solute molecule contains n submolecules and occupies n lattice sites in such a way that successive submolecules occupy closest neighbour sites on the lattice. In this latter case the vapour-pressure equations are

These equations are used to determine the osmotic pressure of solutions of long-chain polymers, and it is found that in the region of osmotic interest, the osmotic pressure is given by an equation of the form

where c g. of solute per 100 c.c. of solution is the concentration. It is shown that this equation can be written approximately

which is the quadratic relation which has usually been fitted to osmotic measurements. To this approximation π/c plotted as a function of c gives a straight line of which the intercept on the π/c axis determines the molecular weight of the polymer molecule and the gradient determines the number of submolecules in each polymer molecule.

Since the above-named paper was published my attention has been called, by Mr A. F. Crossley, to the fact that a solution for Rutherford's ‘Case 1’ had been given by H. Jeffreys in his Operational methods in mathematical physics (Cambridge Tracts in Mathematics), p. 35. This solution is very neat and direct. As, however, the method depends on the use of inverse operational factors, such as where it is less elementary in character than that followed in my paper.