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.

When a stream is flowing along a wall against an adverse pressure gradient, the skin friction at the wall decreases to zero, at which point or at least very near it the flow seems to leave the surface altogether. Between the main stream and the wall there may be a slow back-flow. Upstream of separation the influence of the wall on the flow is satisfactorily explained by the boundary-layer theory, which among other things presupposes that this influence is confined to a narrow region near the wall. Such a theory is clearly inadequate downstream, however, because this layer rapidly increases in width and, further, the main stream is seriously affected, and consideration of the Navier-Stokes equations would seem to be necessary before a full theoretical explanation of separation could be given. Apart from a note by Dean (2) on the relations between the pressure and velocity at separation, there has, however, been no relevant attack on these equations.

The object of this note is to indicate a numerical method for finding periodic solutions of a number of important problems in conduction of heat in which the boundary conditions are periodic in the time and may be linear or non-linear. One example is that of a circular cylinder which is heated by friction along the generators through a rotating arc of its circumference, the remainder of the surface being kept at constant temperature; here the boundary conditions are linear but mixed. Another example, which will be discussed in detail below, is that of the variation of the surface temperature of the moon during a lunation; in this case the boundary condition is non-linear. In all cases the thermal properties of the solid will be assumed to be independent of temperature. Only the semi-infinite solid will be considered here, though the method applies equally well to other cases.

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.

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set

This note is concerned with the problem of the slow motions of an inviscid, incompressible rotating fluid, and in particular with the motion of a sphere along the axis of rotation. This problem was studied recently by Stewartson (2), who overcomes the principal mathematical difficulty, viz. that of formulating the problem in a coordinate system in which the appropriate differential equation can be solved simply and in which the sphere is a coordinate surface, by a very elegant transformation of independent variables. Stewartson, however, uses inappropriate initial conditions. It is the purpose of this note to discuss the question of initial conditions in the light of results previously obtained by the writer (1).

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 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that

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.