It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.

In this note we wish to study some properties of a square matrix of order n and of rank r,

whose coaxial minors |M| of certain orders, or some of them, are known to be zero. Our main results will lie in two directions, according as the vanishing minors are, within certain limits, of an arbitrary order (Part II) or of two consecutive orders only (Part III).

The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s if

This is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.

In a recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.

In this paper I give some operational representations, according to the Carson-Heaviside Calculus, which, as far as I know, are new; and I deduce some properties of the functions thus introduced.

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

In a former paper published in these Proceedings it was shown that an integral function of order less than 1 cannot have any asymptotic periods, and it was suggested that a function of order 1 can have at most a set Kω(k=±1, ±2, …). This was subsequently found to be the case. Meromorphic functions for which K, the exponent of convergence of the poles, is less than ρ, the order, behave in many ways like integral functions, so we should expect that (i) if 0≦κ<ρ1 there should be no asymptotic periods, (ii) 0≦κ<ρ1 either none or else a single sequence kω(k = ±1, ±2, …). It will be shown that this is so.

In the paper, “Sul sistema di tre forme ternarie quadratiche,” Ciamberlini has derived the complete irreducible system of concomitants for three ternary quadratics and has given a short treatment of their geometrical interpretations. Among the concomitants is the invariant (abc)2 which is symmetrical and linear in the coefficients of each quadratic. The purpose of this note is to give a geometrical interpretation of the invariant, and to extend the result for symmetrical invariants of forms in higher dimensions.

The following is a direct proof of a theorem by Zia-ud-Din.

Let {ν} ≡ {ν1, ν2, …, νp)} be any S-function of weight r + s such that

in the alternant denoted in the theorem by A(αβγ….). Let {μ} be an S-function of weight s, equal to and let {λ} be an S-function of weight r, such that {λ}δ(x1, …,xp) is obtained with coefficient gλμν by diminishing the indices in the alternant {ν}δ(x1, …,xp) according to the theorem.