In the previous Paper I, the Variational Principle has been presented as a principle of natural transformation between conserved quantities and infinitesimal in variances, which preserves the geometrical character of the various standard objects under point transformations. So as to establish the theory as easily as possible, Paper I (2) confined itself to the simple case of autonomous dynamical systems, and their 2n dimensional space. The aim of the present paper is to extend the work to the 2n + 1 dimensional theory, in which time is included as an extra variable, and time dependent systems can be handled. It will be seen that within a suitable framework the extension is straightforward, and there are some rather more satisfactory features of the 2n + 1 dimensional theory - even for autonomous systems when included in the new formalism - than in the previous 2n dimensional theory. The exposition will now be rather concise; readers are requested to refer to Paper I for fuller discussion, particularly of the underlying ideas.

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that either

or else

(ii) δ(x)* μn, are mutually singular for all positive integers n.

The sine-Gordon (SG) equation uxt = sin u arises from many branches of physics, and now is one of the most important equations in soliton theory. There have been many works concerning its soliton solutions, Backhand transformations, symmetries and conservation laws and other properties. In this paper we prove that every polynomial symmetry of the SG equation is Hamiltonian, that is, takes the form of D-l δh/δu.

I. J. GOOD. ‘A new formula for cumulants.’

Henryk Minc has pointed out in correspondence that Equation (12), which was a corollary of Theorem 2, does not agree with the computed values of Schur's matrix ω(rs) given by Graham and Lehmer (1976). In the course of the ‘proof’ I made the error of assuming that , overlooking that s could be a multiple of m, so that the sum can also equal m. Since this was the only error, the statement of Theorem 2 is correct modulo m. In particular it shows that perm(ωrs) ≡ 0 (mod m) but this result is much weaker than the results obtained by Graham and Lehmer.

Theorem 1 is unaffected by this error, and moreover has since been proved again by Groeneveld and van Kampen (1977).

Recent papers by Atkinson and Reuter(1), Brown and Carr(5), and Barbour(2) have proved several important results concerning wave solutions of the usual deterministic model for the spatial spread of an epidemic, such as measles or influenza. In particular Atkinson and Reuter showed that non-trivial wave solutions, in a population along a line, only exist provided the contact distribution function has an exponentially bounded tail, and that the speed of propagation c must be at least some critical value c0. They constructed a solution for c > c0, which Barbour later showed to be the unique solution, modulo translation, at that speed. Brown and Carr showed that a solution was also possible at speed c0, though it has not been possible to show uniqueness at this speed.

The existence of non-homogeneous modes of deformation in an equibiaxially stretched elastic half-space is investigated. The surface of the half-space is assumed to be traction-free.

It is shown that three different eigenmodal deformations are possible, according as the cube of the critical stretch is greater than, less than, or equal to ¾.

For the class of strain-energy functions due to Ogden (9), necessary conditions for bifurcation are deduced-for equibiaxial tension. Results are given and discussed for single-term, as well as two- and three-term Ogden strain-energy functions, for equibiaxial tension and compression.

R. A. Bryce and John Cossey. ‘Subgroup closed Fitting classes.’

The proof of the main theorem (Theorem 1.1) in our paper (1) is incomplete. The entire page 203 does not establish what it purports to, the error lying in the assumption in the last paragraph that the group P(F0/φF0) R has nilpotent length three precisely. However, it could be of nilpotent length two, indeed in particular examples is so. The technique used there arose as a generalization of an ad hoc argument.

We discovered the error in attempting to extend the proof of Theorem 1.1, and in (2) we have indeed succeeded in removing the nilpotent length restriction. Page 203 of (1) must be replaced by a special case of Lemma 7.2 of (2).

Some orthogonal sets of Bessel functions of real order v are identified using the equation Δ2u = utt of the vibrating plate. Our main concern is with the L2 completeness of such sets, and we prove that the well known ‘clamped edge’ type is complete for v > -1, thus completing a result of E. Dahlberg. We also study a very closely related set and show that it needs an extra (non-Bessel) function for completeness.

Our method for proving the completeness is based on one given by H. Hochstadt in connection with Dini functions. We have found it necessary to reorganize Hoch-stadt's method and correct some errors contained in it.

Certain isolated values of v require special attention and we treat these by subjecting the Dalzell completeness criterion to a continuity argument.

Leonard E. Dor. ‘Some inequalities for martingales and applications to the study of L1.’

The author wishes to point out that his affiliation when he prepared the paper should have been included. It was ‘Wayne State University, Detroit, Michigan 48202, and The University of Illinois, Urbana, Illinois 61801.’