Whitham's exact averaged variational principle is found by a new method. Comparisons are drawn between this method and the method advanced by Whitham. Further results about the Euler equations from this variational principle are obtained.

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniform supercritical stream (i.e. the Froude number , where c is the speed of the stream, h is its depth, and g is the gravity constant).

We also deduce a number of inequalities relating F2 to a/h, where a is the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 < F2 < 2.

In this paper we derive the Lorentz-Dirac equation of motion for a charged particle moving in an external electromagnetic field. We use Maxwell's electromagnetic field equations together with the assumptions (1) that all fields are retarded and (2) that the 4-force acting on the charged particle is a Lorentz 4-force. To define the self-field on the world-line of the charge we utilize a contour integral representation for the field due to A. W. Conway. This by-passes the need to define an ‘average field’. In an appendix the case of a scalar field is briefly discussed.

A solution which is exact within the framework of the classical theory of elasticity is obtained for the title problem assuming that the half space is homogeneous and isotropic, and that the die indents at a constant rate. If the shape of the die and the elastic medium are specified, the rate of indentation uniquely determines the outward speed of the edge of the expanding contact zone. The magnitude of this speed, relative to the speeds of the dilatational, rotational and Rayleigh waves in the elastic medium, determines which of four possible characteristic transient stress states will occur. Each of the four ranges of contact speed is solved by the method of rotational superposition of self-similar potentials which is briefly described in the Appendix.

The simply supported rhombic plate under transverse load has received extensive attention from elasticians, applied mathematicians and engineers. All known solutions are based on approximate procedures. Now, an exact solution in a fast converging explicit series form is derived for this problem, by applying Stevenson's tentative approach with complex variables. Numerical values for the central deflexion and moments are obtained for various corner angles. The present solution provides a basis for assessing the accuracy of approximate methods for analysing problems of skew plates or domains.

The percolation probabilities of the face-centred cubic and close-packed hexagonal lattices are found not to be identical; the identity of their critical percolation probabilities remains an open question. Other pairs of closely similar lattices are discussed.

(1) Proc. Cambridge Philos. Soc. 73 (1973), page 173

Dr E. Leibowitz has called my attention to a mistake in my paper on isotropic tensors. (1) On p. 173, if 12 is annihilated as stated and then we annihilate l13 by the same method, a new contribution to l12 appears. The correct method is to choose α so as to annihilate l21. Then a suitable rotation on the 13 plane removes l31 without re-introducing l21. The rest of the argument stands.