Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.

A “roughness parameter”, first used by the author in 1953 (Feather 1953 b) has been re-calculated for 348 points on the mass surface. Systematic features are identified in relation to the variation of this parameter with charge number (Z) and isotopie number (D). In the region of small Z these regularities provide evidence for the persistence of some degree of alpha-unit structure at least as far as Ca. In the region of greater Z (20 < Z < 50) they provide evidence for neutron-proton interactions among the last-added nucleons. Overall, they indicate that the “residuals” characterizing the various semi-empirical mass equations currently in use very probably arise in large part from sub-shell effects which it would be impracticable to attempt to include in the equations.

Many results concerning real orthogonal matrices have their counterparts in the theory of orthogonal Boolean matrices. In particular, the analogues for Boolean matrices of certain theorems due to Kronecker are established. The structure of the group of orthogonal Boolean matrices of order n is determined in the case where the underlying Boolean algebra is finite.

An investigation has been made of the condensation nuclei created by an electric field in a N2 – H2O mixture. These nuclei are distinguished by the fact that they induce condensation in a vapour which is only 4 per cent supersaturated.

An explanation of these phenomena is found in the presence of nitrogen dioxide vapour, one of the products of reactions induced by the field, exerting a small pressure PNO2<1O−6 mm. Hg.

The observations are consistent with the assumption that the nuclei are created in the reaction, 2NO2 + H2O ⇌ HNO2 + HNO3. It is believed that the reason for the requirement of a more than critically supersaturated vapour is that this must be the condition for the nuclei forming reaction to proceed.

Once the nuclei have been created, any additional quantity of NO2 collected from the vapour forms acid molecules which promote condensation from a vapour which is not necessarily supersaturated.

Drops formed by these nuclei contain a significant quantity of HNO2 + HNO3, so that, unlike drops of pure water, they are stable against reevaporation in a vapour, the relative humidity of which is <100 percent.

A matrix analysis is derived, using the concept of finite elements, for beam-columns of continuously varying cross-section and “small” initial curvature subject to tangential and normal forces acting at discrete points along the centroidal axis.

The relationships between loads applied to the ends of the member and their corresponding deformations are established in the form of stiffness matrices, which are the basis of the Equilibrium Method of Analysis of linear or non-linear elastic plane frames composed of such membeis. In addition, the end loads induced by the tangential and normal forces for various types of boundary conditions are obtained.

Finally, it is shown that the buckling load may be calculated by the determination of the lowest eigenvalue of a certain matrix, provided there is no lateial translation of the ends of the member.