Stokes's stream-function for an axisymmetrical irrotational motion of perfect incompressible liquid, bounded by a plane [1], or by a sphere [2], has been expressed in terms of the stream-function for the motion in an unbounded liquid. Corresponding results are found in this note for the slow steady motion of a viscous liquid, when the motion in unbounded liquid is irrotational.

In 1935 van der Corput, in connection with his work on distribution functions, was led to the following conjecture which expresses the fact that no sequence can, in a certain sense, be too evenly distributed.

Let f = f(x, y, z) be a positive definite form of the type

where x, y, z are integral valued variables, and the coefficients a, …, t are integers whose highest common factor is 1. As the determinant of such a form may be fractional, I define

and

thus — C is the discriminant of the binary form f(x, y, 0), and the necessary and sufficient condition for f to be positive definite is that a > 0, C > 0, and d > 0.

Among Schubert's many experiments in the application of a symbolic calculus to problems of enumerative geometry, some special attention is due to his long memoir entitled “Anzahlgeometrische Behandlung des Dreiecks” [1]. For one thing, he is dealing here with a simple, though not elementary, kind of geometric variable, the triangle in a fixed plane, so that the paper gives a clear insight into his general method; and, for another, there is contained in this paper, as was recently suggested by Freudenthal ([2], p. 19), an apparently miraculous device, the introduction of “infinitesimal triangles”, which we can now recognize (§4) as having had the effect of desingularizing the triangle domain in which the calculus was to operate. The principal target of Schubert's investigations was the discovery of Bézout-type formulae for the number of triangles common to two algebraic systems Σr and Σ6-r (r = 1, 2, 3) of complementary dimensions, the systems being supposed to intersect in only a finite number of triangles, and the multiplicities of these triangles being assumed to be suitably defined. His systems, also, had to be “normal” i.e. they could only contain such sub-systems of degenerate triangles as were of the dimensions he regarded as normal. He found, by his methods, that “normal” system Σ1 and Σ5 are each characterized (in so far as intersection numbers are concerned) by 7 projective characters, systems Σ2 and Σ4 by 17 such characters, and systems Σ3 by 22 such characters.

The object of this paper is to solve the Saint-Venant torsion problem for those cross-sections with inclusions, which are such that the z-plane boundaries involved can be mapped into concentric circles in a complex ζ-plane by the transformation

with z´(ζ) ≠ 0 or ∞ within the cross-section. We shall consider both solid and hollow inclusions having different elastic rigidities μ. In the case of the solid inclusion we have to restrict the coefficients as to be zero for all negative s, but it is an advantage to leave this restriction to the end of the analysis, since the forms of certain coefficients in the two cases differ only in this respect.

The formula to be proved is

where R(z)>0.

The first formula to be proved is

where p ≧ q + 1, | amp z | < л, R(k±n + αr)>0, r = l, 2, …, p. For other values of p and q the result is valid if the integral is convergent. A second formula is given in § 3.

The following formulae are required in the proof:

where R(z);>0, (1);

where R(α)>0, | amp z | < л, (2);

where the contour starts from -∞ on the ξ-axis, passes round the origin in the positive direction, and ends at -∞ on the ξ-axis, the initial value of amp ζ being - л, (3).

Consider the elliptic integral of the first kind

or, alternatively,

where x = sin ϕ. It is customary in the theory of elliptic integrals to let k = sin α, k' = cos α, sot that k'2 + k2 = 1. For convenience we shall also introduce the parameter

In this paper I construct a reduction formula for the integral

the formula connects any three consecutive members of a set I0, I1, I2, …, Im. We regard m as given, and, in order to avoid wasting time over trivialities, we postulate that the constants a, α, β, γ δ (which are not restricted to be real) have real parts large enough to ensure (i) that the integrals In under consideration and the integrals related to them which will be introduced subsequently are all absolutely convergent, and (ii) that, in all the partial integrations which will be effected, the integrated parts vanish at both limits.

The problem discussed in this paper is a rather specialised problem associated with a particular random walk with reflecting barriers. Such processes are discussed in particular by Feller (2). It happens that probabilities at any time associated with the system here discussed are relatively easy to determine and it is this fact which makes the given solution possible. This problem arose in certain investigations concerned with computing machines carried out by Mr. A. E. Roy of the Astronomy Department in this University and I wish to thank Mr. Roy for bringing it to my notice.

The formulae to be proved are

where m is a positive integer, p ≧ q + 1, R(mar + k) > 0, r = 1, 2, …, p, and | amp z |ππ. For other values of p and q the result holds if the integral is convergent.

We continue our studies (2, 3, 4, 5) of the algebraic, geometric, and analytical similarities and contrasts between Boolean algebras and the real field. In this note we contrast the convergence of series in set algebras with that in the real field.

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:

choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that if

is an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequence

may be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequence

is exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

In a paper published recently in this Journal, Watson obtained a reduction formul for the integral

He showed that this reduction formula could be written in the form:

If

then

where

and 2σ = α + β + γ + δ, 2θ3 = α − β − γ + δ.

This functional equation for Dedekind's

has been established in many different ways; however, it seems that the following proof, a straightforward application of the theorem of residues, has not been observed before. Since

it suffices to prove that

It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfying

It is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy