The star diagram of a diagram [λ] was first obtained by Robinson (5) who denned and studied it, thus giving a unique characterization of the hook structure of [λ]. Staal (6) continued Robinson's work and obtained some important results. Recently, Littlewood (2) published a paper on the modular representations of the symmetric group in which he defined ‘congruent partitions’, ‘p-quotients’, etc., relating to the diagram [λ]. Littlewood's ideas, though superficially different from those of Robinson and Staal, are in essence equivalent to them.

Let C be a bounded closed convex body in n dimensions, symmetric about the origin. Any lattice Λ containing the origin but no other interior point of C is called admissible. There is a positive lower bound Δ(C) for the determinants of admissible lattices (since the origin is inside C); and any admissible lattice with determinant Δ(C) is called critical. Suppose that Λ is any admissible lattice, with determinant d(Λ). We may define A by a fixed set of generating points Li (i = 1,2, …, n); and we shall say that a lattice Λ′ lies in a small neighbourhood of Λ if Λ′ can be generated by a set of points L′i (i = 1,2, …, n) each of which lies in a small neighbourhood of the corresponding Li. We shall call Λ extremal if in a sufficiently small neighbourhood of Λ there are no admissible lattices Λ′ with d(Λ′) < d(Λ). Thus all critical lattices are extremal.

Nicholson (3) stated that the Legendre functions of the second kind, Qn(x), are orthogonal between − 1 and 1. This is incorrect in general, and one purpose of this paper is to give the correct value of

when m ╪ n. Nicholson's value of the integral when m = n, namely,

is correct.

Theorem. Let K be a bounded star-bodyin (n + 1)-dimensional space symmetric about the origin and of volume V < 2ζ(n +1). Then there is a lattice of unit determinant with a point at the origin and having no other points in K. (See (1, 2,3,4).)

The ordinary theory of stability of plane parallel flows is considerably simplified by a result due to Squire (2) which says that if a velocity profile becomes unstable to a small three-dimensional disturbance at a given Reynolds number, then it will become unstable to a small two-dimensional disturbance at a lower Reynolds number. This result enables us to restrict investigation of the stability to the cases of two-dimensional disturbances.

The circle theorem of Milne-Thomson(1) connecting the complex potential in a two-dimensional irrotational flow about a circular cylinder with that of the flow when the cylinder is absent has a three-dimensional counterpart in the result due to Weiss (3) for the perturbed velocity potential in an unlimited irrotational flow when the rigid spherical boundary r = a is inserted.