A multi-partite number of orders j is a j dimensional vector, the components of which are non-negative rational integers. A partition of (n1, n2, …, nj) is a solution of the vector equation

in multi-partite numbers other than (0, 0, …, 0). Two partitions, which differ only in the order of the multi-partite numbers on the left-hand side of (1), are regarded as identical. We denote by P1(n1, …, nj) the number of different partitions of (n1 …, nj) and by p2(n1, …, nj) the number of those partitions in which no part has a zero component. Also, we write P3(n1, …, nj) for the number of partitions of (n1; …, nj) into different parts and p4(n1, …, nj) for the number of partitions into different parts none of which has a zero component.

Whittaker has shown that a general solution of Laplace's equation, ∇2V = 0, may be expressed in the form

Since the harmonic property of a function is in no way dependent upon any particular set of axes it follows that the same solution must be capable of being expressed in the form

where X, Y, Z are any second set of rectangular coordinates.

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae

1. Let the sides AB, AC subtend equal angles λ at P.

Take ABC as the triangle of reference and denote AP, BP, CP by x, y, z. Then we have, using trilinears,

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

Definition. When two conics intersect each other at two points in such a manner that the tangents and normals of the one become the normals and tangents of the other, they may be said to cut each other orthogonally.

Theorem. 1st, A given conic can be cut at every point on it by two conics which are orthogonal to it; 2nd, every conic orthogonal to a given conic passes through two fixed points on the axis of the given conic.

Let E be a separated locally convex barrelled space with continuous dual E′ and algebraic dual E* and let M be a subspace of E* with and dim Robertson, Tweddle and Yeomans have recently considered the question of barrelledness under the Mackey topology τ(E,E' + M) when E is given to be barrelled under its original topology τ(E,E') [5], [6], [7].

An extensive investigation into the geometry associated with the binary cubic form as representing a triad of points on a twisted cubic curve has been carried out by Sturm. The object of the present note is to derive two further geometrical properties, one associated with the twisted cubic, and the other with the rational plane cubic curve.

We show that if a periodic residually-finite group G has a 2-element with finite centralizer then G is locally finite.

1. Let

be an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notations

describing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).

Based on the theory of p-supersoluble and supersoluble groups, a prime-number parametrized family of canonical characteristic subgroups Γp(G) and their intersection Γ(G) is introduced in every finite group G and some of its properties are studied. Special interest is dedicated to an elementwise description of the largest p-nilpotent normal subgroup of Γp(G) and of the Fitting subgroup of Γ(G).

In the paper referred to above (see [1]), the following corrections should be made.

(1) Hypothesis 2.2 should read:

(2) The set E⊂ℝn should be defined as

(3) F0(υ)=υ+x should read F(υ)=υ–x.

(4) The unique solution of F0(υ)=0 is υ=x (not υ= –x as printed).

(5) Page 345, line 14:

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kind

and the triple series equations of the second kind

where f, f1, g, g1h and h1 are all known functions,

is the Jacobi polynomial (3).

Let S be a finite semigroup. Consider the set p(S) of all elements of S which can be represented as a product of all the elements of S in some order. It is shown that p(S) is contained in the minimal ideal M of S and intersects each maximal subgroup H of M in essentially the same way. The main result shows that p(S) intersects H in a union of cosets of H′.

The numerical solution of Integral equations with variable upper limits has been investigated by Professor Whittaker. In this investigation the nucleus, supposed to be given numerically by a table of single entry, is replaced by an approximate expression consisting of a finite number of terms, each term involving an exponential or simply a power of the variable, and then the solution is found as an analytical expression from which its numerical values may be computed. The numerical solution of integral equations with fixed limits has been discussed by H. Bateman.

It is evident in the first place as is pointed out by Steiner that the conic will be a parabola if, and cannot be a parabola unless the point at infinity on one range correspond to the point at infinity on the other, that is, the two ranges must be similar. This is the converse of the well-known proposition that a movable tangent to a parabola divides two fixed tangents similarly.