Among the many formulæ which show special relations existing between the circular functions and the Bessel-Function Jn(x), when n is half an odd integer, there is one due to Lommel

The homogeneous real linear transformation in n variables is such that, when these variables are used as a set of mutually rectangular coordinates, an n-dimensional sphere is transformed into an n-dimensional ellipsoid; n mutually rectangular radii of the sphere become the n, mutually rectangular, principal radii of the ellipsoid. When these principal radii have not been rotated from their original directions, the transformation is said to be pure, or irrotational. Since these radii are necessarily real, the roots of the n-ic for the determination of the n principal elongations are necessarily real.

In this section we extend the definition of an E-set, so that it includes sets of the type

where the only restriction on the Ei is that they be non-singular. We now consider matrices of the type

where each ei takes independently the values 0, 1, …, n − 1, while the a(ei) are either complex numbers or else matrices of order r, the product a(ei) E(ei), in the latter case, being interpreted as the direct product of the two matrices a(ei) and E(ei).

The following is a simple geometrical demonstration of the well-known theorem that, if matter be distributed over a sphere with a surface-density (i.e., mass per unit of surface) inversely as the cube of the distance from either of two points which are the inversions of each other with respect to the sphere, it will act upon all external masses as if it were collected at the interior point:—and upon all internal masses as if a definite multiple of its mass were concentrated at the exterior point.

General Construction for Refracted Ray. The two-circle method of finding the direction of a ray refracted at a plane surface is very old, but seems to be now almost forgotten. It is particularly convenient when the refraction of several rays is to be determined, and its application to the case of a prism is especially elegant and leads to a simple self-contained proof of the condition for minimum deviation.

Let $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.

We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.

AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

In [1], R. L. Goodstein has extended some well-known theorems on functions and equations in a Boolean algebra to the case of a distributive lattice L with 0 and 1. The purpose of this paper is to prove that most of Goodstein's theorems, as well as some additional results, are still valid in the case when L is not required to have least and greatest elements.

We shall extend some of the results of (7) to the case of multiple alleles, our primary concern being that of polyploidy combined with multiple alleles. Generalisations often tend to make the computations more involved as is expected. Fortunately here, the attempt to generalise has led to a new method which not only handles the case of multiple alleles, but is an improvement over the method used in (7) for the special case of polyploidy with two alleles. This method which consists essentially of expressing certain elements of the algebra in a so-called “ factored ” form, gives greater insight into the structure of a polyploidy algebra, and avoids a great deal of the computation with binomial coefficients, e.g. see (7), p. 46.

Sr denotes an infinite space of r dimensions. Such a space is divided into two regions by a Sr−1. An infinite line is divided into n + 1 regions by n points. An infinite plane or other surface topically equivalent to it is divided into two regions by an infinite line.

§ 1. Corresponding to any partition

which we denote by [a1, . …, ak] or briefly by [a], of the integer n, we can construct a shape which has a1 spaces in the first row, a2 in the second row, . …, ak in the kth and last row. Thus the shape corresponding to the partition [5, 3, 3, 2] of 13 has the form:

The properties of the functions Pn, Qn may also be determined very briefly as follows:—

Let u be a function of x; then by integration by parts we have