Many proofs of the Binomial Theorem have been given, and the proof which I give in this note has not the slightest claim to be considered new. My only object in giving it is to call attention to the fact that it depends merely on the rule of integration by parts, and provides a form of the remainder that in many applications is much simpler than any of the forms associated with Taylor's Theorem. Of its usefulness in this latter respect I give an example from the asymptotic expression for a Bessel function.

Let P be a semilattice. In (5), a ring T is called a supplementary semilattice sum of subringsTα (α∈P) if the following conditions hold: TαTβ⊆Tαβ for all α,β∈P, and for each α∈P. Thus, as an abelian group, T is a direct sum of the additive subgroups Tα (α∈P), and the multiplicative structure of T is strongly influenced by the semilattice P. Properties of these rings have been studied extensively in (2), (3), (5), and (6).

The condition that the normals at the points whose eccentric angles are α, β, γ shall be concurrent is

The following method of establishing this result, as compared with those given in works on Concis, is direct, and also has the advantage of simplicity, by first proving the Trigonometrical identity

which may be simply done by multiplying both sides of the wellknown identity

when it will be easily found that the product in the first member reduces to that in (A) with opposite sign.

It is characteristic for the development of many processes that in certain moments they change their state in jumps. Systems with impulse effect provide an adequate mathematical model of such processes. The investigation of these systems begins with the paper of Millman and Myshkis [7] and afterwards the number of publications dedicated to this problem rapidly increases.

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

In the present paper I propose to investigate the fundamental geometrical properties of the Apolar Locus of two tetrads of points in a plane.

The Apolar Locus of two tetrads of points K, L, M, N and P, Q, R, S is defined in the locus of the point X, moving so that the pencils X[K, L, M, N] and X[P, Q, R, S] are apolar.

Consider a non-holonomic dynamical system specified by the N coordinates qi, and kinetic energy defined by

where amn are functions of qi and the dot denotes differentiation with respect to the time t.

In a previous paper the author has employed the expansion

where

and

to establish the theorem that, if −π/2<amp γ<π/2, the function Ps/Ts+1 remains finite as γ→∞. This theorem is valid provided that z is not real and ≧ 1.

The boundary value problem

is studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.

This boundary value problem arises in the search of positive radially symmetric solutions to

where Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

In a paper recently read before this Society, Mr E. Blades obtained a general formula for spheroidal harmonics in the form of the general solution of Laplace's equation given by Professor Whittaker,

If spheroidal coordinates r, θ, φ are defined by

the result obtained is

The theorems given here deal with the efficiency, mutual consistency, and regularity of γ-transformations.

§1. Theorem 1. The necessary and sufficient conditions that γ-matrices are efficient for all divergent series whose partial sums are bounded, are

where Δgk(a)≡gk(a)−gk+1(a).

(1). Let ABC be the given triangle; A′BC, B′CA, C′BA triangles described externally on its sides, and let the angles of these triangles be A′BC=μ1, A′CB=v1, B′AC = λ2, B′CA = v2 C′AB = λ3, C′BA = μ3, (Fig. 1).

If h is an outer function in H1 then it is shown that h = (q1 + q2)g where both q1 and q2 are inner functions with Im almost everywhere, and g is a strong outer function (equivalently, g/∥g∥1 is an exposed point of the unit ball of H1). If q1 + q2 is nonconstant then such an h is not strongly outer. Moreover a sum of two inner functions is studied.

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

The result of Ballantine [1] to the effect that a singular matrix A is a product of k idempotent matrices if and only if the rank of I – A does not exceed k times the nullity of A is generalized to endomorphisms of a class of independence algebras.

It is an obvious remark that the Mathieu functions, being the harmonic functions of the elliptic cylinder, must be closely related to the Bessel functions, the harmonic functions of the circular cylinder. Reference has been made to some aspects of this relationship in two earlier communications, to which the present paper may be regarded as a sequel.