In our article [8] we examined asymptotic mean square stability for linear retarded f.d.e.'s which are perturbed by white noise. It is shown in [8] and [10] that if the deterministic linear retarded f.d.e. is asymptotically stable, then so is the perturbed stochastic f.d.e.

Laguerre has shown that if be expanded in ascending powers of t,

where fn(x) is a polynomial of degree n, satisfying the differential equation

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered was

where is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-element

was that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

Let α, β, γ be the known trilinear coordinates (actual lengths of perpendiculars) of the centre of a conic inscribed in the triangle of reference ABC.

In the Euclidean theory of areas, where convex polygons alone are considered, there is no question as to the sign of an area. The element of area is the rectangle, and an area is signless, or always positive.

It is well-known that if v≧–½ and

then ø (x) is said to be self-reciprocal in the Hankel transform and may be described as Rv. If v = ±½(1) reduces to the Fourier sine or cosine transform. Functions of these two classes may be described as Rg and Re.

In a list of pairs of reciprocal functions, G. A. Campbell gives the example that

is Rc, but there does not seem to be any explicit reference to this function in the literature. It suggests that there is a corresponding function which is Rv.

W. H. Cornish (2) has investigated congruences on pseudo-complemented distributive lattices and has identified those ideals (resp. filters) that are congruence kernels (resp. cokernels). In this paper we show that many of the principal results concerning congruence kernels and cokernels hold in a semilattice and therefore do not depend on distributivity, nor on the existence of unions.

Three main aws regulate the treatment of ordinary algebraic quantities. These are the Associative Law, the Distributive Law, and the Commutative Law. If a, b, c, … , represent quantities dealt with in the algebra, the associative law of multiplication asserts that a(bc)=(ab)c, where the brackets have the usual meaning that the quantity within them is to be regarded as a single quantity: the distributive law of multiplication asserts that (a + b)(c + d)=ac + bc + ad + bd: and the commutative law gives ab = ba. With regard to addition, the associative law asserts that (a + b) + c = a +(b + c): and the commutative law gives a + b = b + a.

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—

If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.