A well-known theorem in group theory [(8), p. 11, Satz 3] asserts that, when H is a subgroup of finite index in a group G, there exists a system of common representatives of the right cosets and the left cosets of H in G. Various proofs and generalisations, mainly involving combinatorial rather than grouptheoretical ideas, are known, and an excellent account of the subject is to be found in Chapter 5 of Ryser's book (6), where references to the literature are given. The purpose of the present paper is to use group-theoretical ideas to prove theorems of a similar nature. The motivation for this work comes from the theory of Hecke operators, and one of the main objects is to provide a simple proof of a result given by Petersson (4), which is needed in order to prove the normality of these operators.

In the following pages we give a complete system of projective concomitants for any number of linear complexes and one quadric, in a quaternary field. The investigation follows the earlier1 paper on mixed quaternary forms, in which the corresponding system, omitting the quadric, was given. Such a system, for one quadric and one linear complex was given by Weitzenböck who used complex symbols. A detailed investigation of the case of two linear complexes and a quadric, together with their geometry, has been given elsewhere.

We use the Mittag-Leffler partial fractions expansion of jv + 1(x)/Jv(x) to give simple proofs of some recent results due to S. H. Lehnigk concerning the number of positive roots of the equation ( −Br2 + A + q)Iq(r) + rIq,+ 1(r)=0, where A is real, B>0 and q>−1.

A friend, Professor Smith of Sackville, New Brunswick, has called my attention to the fact that the term “Wallace's Line,” as defined in the Century Dictionary, is employed by other scientists than mathematicians. In these days, when increasing attention is paid to the nomenclature of lines and curves, it may be of interest to add here a brief quotation from the dictionary: “Wallace's Line [so named after Alfred R. Wallace, who defined it] in Zoögeography, a line assumed to separate the Indomalayan from the Austromalyan Zoological region or faunal area. It passes between Borneo and Celebro, through the Strait of Macassar. Southward between Bali and Lombok, northeastward between Mindanao and Gilolo. This line divides the shallow waters of the Indomalayan region from the much deeper Austromalayan seas; and the character of the fauna is quite different on the two sides of it.”

ABCDEF is any cyclic hexagon;

AB, DE meet in G,

BC, EF „ „ H,

CD, FA „ „ K; then G, H, K are in a straight line.

Draw KLNM parallel to BC and produce DE, EF, AB to meet it in I, N, M. Join DA, DM and let BC, DE meet in P.

In some biological problems, a parasite-host system is immersed in a toxic solution in order to kill off the parasite while leaving the host as little affected as possible. A problem of this type was considered by Clements and Edelstein (2), who treated both the host and the parasite as cylindrical in shape. In a separate paper Clements (1) considered the corresponding problem where the parasite is spherical and the host cylindrical. In both cases, the concentration of the toxic solution at the boundary is taken as having a constant value, c, and the penetration of the poison into the host and parasite is treated as a linear diffusion problem with an appropriate diffusion coefficient. It is assumed also that the host and the parasite are free of the toxic substance initially. The process is terminated when the average concentration in the parasite reaches a lethal level, τ, and the problem is to see how M, the average concentration in the host, is affected by the choice of c (for a given value of τ).

In a paper published in 1897 it was suggested that the Law of Extensible Minors was applicable to Kronecker's linear relation between the n-line minors of an axisymmetric determinant of the (2n)th order, the concluding sentence of the passage being— “For example, knowing from Kronecker that

if the terms be minors of the axisymmetric determinant

we can at once vouch for the identity

in connection with the axisymmetric determinant

No proof was given, as the matter turned up quite incidentally, and verification of the cases made use of was all that was necessary.

1. Jacobi obtained his well-known formulae by a purely algebraic method, but it was not until H. J. S. Smith had obtained them similarly, but by the use of a more symmetrical notation, that they were put into the form by which they are known today.

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.