The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.

Many theories have been advanced to account for a presumed uniformity in the temperature of the earth in past ages, and one of the most recent is that advanced by Sir John Murray in the Summary Volumes of the Challenger Report. A careful study of the distribution of marine fauna showed the existence of remarkably similar organic forms in the Arctic and Antarctic regions which were entirely unknown in intermediate waters. The existence, also, of ancient corals in Polar seas points to a high Polar temperature at some remote period of the earth's history. These facts combined lead us to consider it probable that in some past age the earth's air temperature was high and uniform.

In 1730 there was published Stirling's Methodus Differentialis, and in it (Prop. VIII., p. 44) he considers the Difference Equation

and shows that it is satisfied by an inverse factorial series

If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,

If we now regard λ as a parameter and z as the independent variable then (1.1) can be written

Supposing, therefore, that a function is of the form then

In this article we prove that in every infinite dimensional separable Fréchet space there is a dense barrelled subspace which is not the inductive limit of Bairehyperplane spaces.

Throughout this note, N will denote a (Left) near-ring with two-sided zero. Definitions of basic concepts can be found in (9).

We prove first that a right ideal I in a d.g. near-ring has a right identity if and only if x ∈ xI for each x ∈I. This enables us to study the structure of regular d.g. near-rings with chain conditions on right annihilators. Specifically we will prove that a regular d.g. near-ring with both the maximum and the minimum conditions on right annihilators is a finite direct sum of near-rings which are either rings of matrices over division rings or non-rings of the form MG(Γ) for a suitable type 2 N-module Γ. Finally we consider the case of maximum condition on N-subgroups.