We show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

Mathematical induction in its simplest form may be stated thus: Suppose there is a set of propositions p0, p1, p2…which are so related that the truth of pn implies the truth of pn+1 then if p0 is true, it follows that all the other propositions of the set are true. For since p0 is true therefore p2 is true, therefore p2 is true, and so on as far as we please.

So far as I am aware, the methods, described below, of finding the attraction of a uniform shell of matter on a particle placed at external or an internal point, are new. They are particular applications of a method of finding the attraction of a thin shell uniform volume density bounded by similar and similarly situated ellipsoidal surfaces (what some have called a homothetic shell, and Thomson and Tait an elliptic homceoidœ) which I explained a paper, on the attraction of ellipsoidal shells and of solid ellipsoids, which was published in the Philosophical Magazine for April 1907. I give here also a short account of the more general problem with some additional notes and remarks. The solution depends on a geometrical theorem of some interest which occurred me in thinking over the problem of the ellipsoidal shell, and its solution by Poisson by a laborious and somewhat difficult process integration (Me'moires de I'Institut, t. xv., 1835).

If the distances (12), (13), (14), (23), (24), (34) between four points 1, 2, 3, 4 on the circumference of a circle be denoted by a, b, c, d, e, f respectively, then a certain relation (A) is known to connect a, b, c, d, e, f. The same four points, however, being points in a plane, there subsists between their mutual distances another relation (B). Now, it occurs to one that from these two relations some deduction ought to be possible regarding the mutual distances of four points on a circumference, and the problem is suggested of making the said deduction.

Let there be given, on an algebraic curve C, of genus p, a linear series and an algebraic series of index v, both without fixed points. The number of groups of r + 1 points which are common to a set of and a set of has been shown by Schubert (1) to be

where d is the number of double points of .

A study of nonselfadjoint algebras of Hilbert space operators was begun by considering special types of such algebras, namely those determined by a commuting family of rank one operators. A first step in this direction was made by Erdos in [1] and is continued more extensively in [2].

A linear algebra of order n, in general non-commutative and non-associative, may be regarded as being determined by the “cubic matrix ”consisting of its n3 constants of multiplication, and conversely. This requires that the n basis elements (units) of the algebra should be specified, and should be given in a definite order. Then the various “transpositions” of the cubic matrix induce corresponding “transpositions” of the algebra, for which a notation is given in §2.

It has been pointed out that, in the paper by H. W. Richmond (Proceedings of the Edinburgh Mathematical Society (2), 6 (1940), 190-191), lines 3 and 4 of page 191 should read as follows:—Now the pencil includes two ruled cubic surfaces, one of them having 8′ as its double line and 7′ for the second line which the generators meet.

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

If on the sides of a triangle ABC equilateral triangles LBC MCA NAB be described externally, AL BM CN are equal and concurrent.

The group of symplectic transformations acts on the unit ball of a Hilbert space. The structure of the orbits has been determined by N. J. Young in [8]. We provide a new proof of this theorem; it is slightly simpler than the original one, and does not involve Brown–Douglas–Fillmore theory. Moreover, the steps followed hopefully throw some additional light on the subject. We rely heavily on previous work of Khatskevich, Shmulyan and Shulman ([5, 6, 7[); the proofs of the results used are included for completeness.

The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equation

and can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,

The aim of this paper is to discuss groups G=HK=HA=KA with a triple factorisation as a product of two subgroups H and K and a nilpotent normal subgroup A. It is of interest to know whether such a group G satisfies some nilpotency or supersolubility condition if H and K satisfy the same condition. A positive answer to this problem is given for certain group classes under the hypothesis that A is prefactorised in G = HK. Some applications of the main theorem are also mentioned.