Helmholtz and most, if not all, subsequent writers on vortex motions have, except in obtaining the fundamental equations, confined themselves to fluid of invariable density.

In the following paper are considered some simple systems of vortices in a compressible fluid. To show that such systems are of considerable importance it is sufficient to refer to the phenomena of cyclonic storms. It may be as well, however, to state that though the vortices are here treated as compressible, the circumstances are still so different from those found in nature that the results obtained could bear only a general resemblance at most to the phenomena of storms.

An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relations

between k generating operations S1, …., Sk of , such that every other relation between S1, …., Sk is an algebraic consequence of (1).

The number of groups for which abstract definitions are actually known is relatively small, but a remarkable feature of the results already obtained is the extreme simplicity of the relations (1) in the case of several groups of quite high order. This fact constitutes an additional incentive to the search for abstract definitions, and many elegant results have doubtless yet to be discovered.

I have recently shown that any two straight lines that are isotomically conjugate with reference to a triangle are the asymptotes of a conic circumscribed about the triangle; and that, conversely, any circumscribed conic has for its asymptotes two such lines. Now there is a one-fold infinity of conies having a given pair of straight lines for asymptotes; and, as there is a two-fold infinity of pairs of isotomic lines in the plane (with reference to a given triangle), it follows that two conditions must subsist among the coefficients in the equation of a conic, in order that its asymptotes may be isotomic lines.

Let

and

be, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.

A concordance classification of links of , p < 1, is given in terms of an algebraically defined group, Φ±, which is closely related to Levine's algebraic knot concordance group. For p=1,Φ_ captures certain obstructions to two component links in S3 being concordant to boundary links, the generalized Sato-Levine invariants defined by Cochran. As a result, purely algebraic proofs of properties of these invariants are derived.

One of the outstanding theorems in the theory of continued fractions is the result described by O. Perron, Die Lehre von den Kettenbrüchen (1912, 1929, 1954–7) as the transformation of Bauer and Muir (for brevity I shall call it the BM theorem); this theorem and limiting cases of it give rise to numerous extremely interesting consequences.

We study systems which in characteristic coordinates have the form

where A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).

The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.

Simple weak converge of the initial data does not imply weak convergence of the solutions.

We characterize regular bisimple rings in terms of some perspectivity conditions on their lattices of principal right ideals. We also show that, if S is the multiplicative subsemigroup generated by all the idempotents of a regular bisimple ring R, then

(i) if R does not have an identity, then S = R and has depth 2;

(ii) if R does have an identity but is not a division ring, then S = {a∈R:a is neither left nor right invertible} ∪ {1} and has depth 3.

In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p-classes of compact operators. The ideas developed there can be used in ℓp-spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions

The object of the following paper is to consider the motion of one or more vortices in a compressible fluid, which is rotating as a whole with uniform angular velocity ω about an axis, taken as axis of z. To save space I shall when possible refer for results to a previous paper in the Proceedings, distinguishing the equations of that paper, Vol. V.; pp. 52–59, by the suffix a.