Lorsqu' on intègre successivement les fonctions

où le cosinus de l'arc α est multiplié par une puissance à exposant entier de l'arc lui-même,

on reconnaît que les intégrales sont toutes comprises dans la formule

où P et Q représentent des polynomes entiers en α que l'on détermine dans chaque cas particulier.

The field-equations of gravitation in Einstein's theory have been solved in the case of an empty space, giving rise to de Sitter's spherical world. In the case of homogeneous matter filling all space, the solution gives Einstein's cylindrical world. The field corresponding to an isolated particle has been obtained by Schwarzchild. He has also obtained a solution for a fluid sphere with uniform density, a problem treated also by Nordström and de Donder. A new solution of the gravitational equations has been obtained in this paper, which corresponds to the field of a heterogeneous fluid sphere, the density at any point being a certain function of the distance of the point from the centre. The law of density is quite simple and such as to give finite density at the centre and gradually diminishing values as the distance from the centre increases, as might be expected of a natural sphere of fluid of large radius. The general problem of the fluid sphere with any arbitrary law of density cannot be solved in exact terms. It will be seen, however, from a theorem obtained in this paper, that the solution depends on a linear differential equation of the second order with variable coefficients involving the density, and thus the laws of density for which the problem admits of exact solution are those for which the above coefficients satisfy the conditions of integrability of the differential equation. An approximate solution for any law of density may be obtained by the method of series.

In recent years, the problem of embedding the projective spaces in Euclidean spaces was studied very much, by different methods. Usually, the negative results on the embedding problem are proved by using suitable homotopy invariants. The best known example of such homotopy invariants is given by the Stiefel–Whitney classes.

We show that if G is a compact abelian group and U is a weakly continuous representation of G by means of isometries on a Banach space X, then holds for each measure µ in reg(M(G)), where π(µ) denotes the generalized convolution operator in B(X) defined by , σ the usual spectrum in B(X), sp(U) the Arveson spectrum of U, the Fourier-Stieltjes transform of µ and reg(M(G)) the largest closed regular subalgebra of the convolution measure algebra M(G) of G. reg(M(G)) contains all the absolutely continuous measures and discrete measures.

The paper is devoted to studying one generalization of Steiner systems S(n, k, l) closely related to packings and coverings of l-tuples by k-tuples of an n-set. One necessary and one sufficient condition for the existence of such designs are obtained.

§1. Since the former paper on this subject was read, Prof. Cantor has published the second volume of his history of Mathematics. This has necessitated various additions to the paper, which can perhaps be best given as an appendix.

On page 413 Prof. Cantor says that the construction of Dürer's pentagon is found in a book called Geometria deutsch, which was lately discovered in the town library at Nürnberg, and gives 1487 as the upper limit to its date. The construction is said to be “mitunverrücktem Zirckel,” the same expression that Schwenter applies to Dürer's solution.

The theorem is

If the straight lines bisecting the angles at the base of a triangle and terminated by the opposite sides be equal, the triangle is isosceles.

This theorem was in the year 1840 communicated by Professor Lehmus of Berlin to Jacob Steiner with a request for a pure geometrical proof of it. The request was complied with at the time, but Steiner's proof was not published till some years later.

The relationship between certain non-associative algebras and the deterministic theory of population genetics was first investigated by Etherington (3)-(8), who defined the concepts of baric, train and special train algebras. Gonshor (10) dealt with, among other topics, algebras corresponding to autopolyploidy, on the assumption that chromosome segregation operated. In this paper [ discuss algebras corresponding to more general systems of inheritance among polyploids, which have been discussed without using algebras by Haldane (11), Geiringer (9), Moran (13) and Seyffert (16). These algebras are special cases of what I have defined as segregation algebras, and mixtures of them. All the algebras corresponding to a fixed ploidy have a relationship which I have called special isotopy. An example shows that algebras arise in other genetic systems which are not isotopic to segregation algebras.

In a recent paper (13), we introduced the class of strongly E-reflexive inversesemigroups. This class was shown to coincide with the class of those inverse semigroups which are semilattices of E-unitary inverse semigroups. In particular, therefore, E-unitary inverse semigroups and semilattices of groups are strongly E-reflexive, and in fact so are subdirect products of these two types of semigroups.

The conception of the integral of one function with respect to another was introduced by Stieltjes in his classical memoir on continued fractions. He denned the integral as

and gave the formula for integration by parts.

We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:

Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.

In the second part of Professor Chrystal's Algebra (Exer. V., No. 4) the following exercise is given:-

If 2xyz-x2-y2-z2+1=0, and x,y,z are all real, then all, or none, of the quantities x,y,z lie between-1 and + 1.

The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.

AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

The known methods of “summing” divergent series, e.g. the means of Cesàro, Riesz, Borel, Lindelöf, Mittag-Leffler are particular cases of the transformation of a sequence (formed from the partial sums) by a T-matrix. An equivalent method is that of the transformation of the series by a γ-matrix, the fundamental properties of which have been proved by Carmichael, Perron and Bosanquet.