This summer there came into my hands a copy of the spring issue of the Mittheilungen der Math. Gesellschaft in Hamburg containing a paper on the “Transformationen der hydrodynamischen Gleichungen mit Berucksichtigung der Reibung.”

On examination, I found embodied in the somewhat lengthy communication practically the following method, which I had entered in my notes three years ago when working at the subject. Thinking that it was bound to have been used earlier, I simply preserved it, as likely to prove useful if I were ever called upon to teach Hydrodynamics.

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namely

Here, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

It is well known that the mean value theorem (MVT) does not, in general, hold for analytic functions. The most familiar example to this effect is f(z) = ez since e2πi−e0≠2πiez0 for any z0∈ℂ. On the other hand, it is easy to show that the MVT holds in ℂ if f(z) is a polynomial of degree at most 2. Thus it is natural to ask what conditions on a function f(z) analytic in a domain D are necessary and sufficient for f(z) to satisfy the MVT in D. This is one of the questions answered in this paper.

Let (K, v) be a complete, rank-1 valued field with valuation ring Rv, and residue field kv. Let vx be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in Rv[x] are such that (i) vx(F(x) – G0(x)H0(x)) > 0, (ii) the leading coefficient of G0(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of vx satisfying vx(A(x)G0(x) + B(x)H0(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G0(x), (c) vx(G(x)–G0(x)) > 0, vx(H(x) – H0(x)) > 0. In this paper, our goal is to prove an analogous result when vx is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of kv.

1. If in a closed area CD a point O is given, and through this point a straight line MN is to be drawn such that the segment MCN so determined may be a minimum or a maximum, this condition is satisfied by drawing through O a chord MN having O for its mid point. Of the two segments which are separated by this chord and which are obtained by rotating the chord round O, the one MDN will be the maximum and the other MCN will be the minimum. The sum of the segments is equal to the whole area of the contour, and consequently if one of them corresponds to the minimum, the complementary segment corresponds to the maximum. The two segments interchange, the one into the other, when the chord MN, turning round the point O, undergoes a displacement equivalent to the angle π.

We are concerned with the stability properties of uniformly closed wedges in C(E) (resp. C+(E)), the real-valued (resp. non-negative) continuous functionson a compact space E, and solve the following problems in this area:

(a) Let A be a closed semi-algebra in C+(E) such that

The following paper was written last summer, and was submitted to Dr Mackay with a view to eliciting his opinion particularly on the curious passage referred to in § 3, and on the remarks contained in § 8. I was not aware of the intention of Mr T. L. Heath to follow up his excellent edition of Apollonius by an edition of Archimedes on similar lines, and when I saw the announcement of his Archimedes in the month of October, I at once concluded that the notes I had made would have been anticipated by him. Since reading his masterly work, however, I am disposed to think there is still sufficient interest in the notes I have written to justify me in laying them before the Society; I therefore submit them in their original form, although I should have omitted certain details had I been acquainted with Mr Heath's work before writing the paper.

Professor Whittaker, in a paper entitled “On Tubes of Electromagnetic Force” {see Proceedings of the Royal Society of Edinburgh, Vol. XLII., Part I. (No 1)}, introduces certain surfaces, which he names calamoids, in connection with an electromagnetic field in the four-dimensional world of space-time. The calamoids consist of “a convariant family of surfaces which when the field is purely electrostatic or purely magnetostatic reduce to the ordinary Faraday tubes of force.” Professor Whittaker, in the paper referred to, also introduces two sets of surfaces, each a covariant family of ∞2 surfaces, one of them named the electropotential surfaces, and the other family the magnetopotential surfaces of the electromagnetic field. The electropotential surfaces and the magnetopotential surfaces are shown to be everywhere absolutely orthogonal. (One member of each family meeting at a point, any line from this point in the one family is orthogonal to every line through the point in the other family). Moreover, a “calamoid, at every one of its points, is half-parallel and half-orthogonal to the electropotential surface which passes through the point, and is also half-parallel and half-orthogonal to the magnetopotential surface which passes through the point.”

Throughout this note it will be assumed that

(i) unless otherwise stated, k; is a positive integer,

(ii) a ≧ 0; m > − 1,

(iii) q ≧ 1; q′ is the conjugate number to q, and is defined by

(iv) the functions φ(w), ψ(w) are defined in [0, ∞) with absolutely continuous kth derivatives in every interval [a, W],

(v) φ(w) is non-negative and unboundedly increasing,

(vi) λ = {λn} is an unboundedly increasing sequence with λ1 > 0.