In the interesting new book by Carslaw and Jaeger, Operational Methods in Applied Mathematics, a point is raised which does not seem to have been explained adequately in this or any other book on this subject that I have seen. This concerns the application of initial conditions to switching problems in electric circuit theory. Reference is made to this deficiency in a review of this book in the Gazette, XXVI, No. 268 (February 1942), pp. 63-64, and the present article is a suggested way of overcoming it.

In this article the method of solution is applied to Bessel’s equation, but it can be used in the same way to solve other linear equations.

If from a point, regarded as a centre of inversion, a radius is drawn on which are two points at distances r1 and r2 from the centre, satisfying the condition r1r2=r2 = constant, the points defined by r1 and r2 are said to be inverses one of the other.

A problem of frequent occurrence in the analysis of engineering structures is the determination of the resultant actions in a uniform bar which is subjected simultaneously to an axial thrust and to lateral loads and terminal couples. This type of problem can be solved by straightforward analytical methods, and well-known formulas for certain cases of loading have been obtained by this means. An alternative method of solution is, however, available by the use of a polar diagram originally due to Ratzersdorfer but discovered independently by H. B. Howard and introduced by him to British engineers. This method is of great, value in obtaining graphical solutions in numerical cases and has been used very widely in drawing offices for this purpose. In addition to its use for the solution of specific problems, however, it offers an attractive method of deducing general formulas and is likely to appeal to the imagination of many students far more than the usual approach. This application of the diagram does not appear to have received the attention that it deserves and the object of this paper is to show its value to the teacher.

The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

In the Gazette, No. 248 (February, 1938), pp. 50-57, Mr. W J Dobbs gave a very interesting and complete descriptive account of the 18 equilateral “Morley” triangles of a given triangle. The article is especially memorable for its diagrams (1-8) and the present reader is asked to have them at hand for reference.

In § 1 of this Note we first investigate some geometrical inequalities connecting a set of n unordered points in a plane. An ellipse is associated with the set of points, and use is made of the invariants of this ellipse under rotation of the coordinate axes. Some inequalities, first obtained by Weitzenböck, are derived and extended. A new geometrical entity, which we call the mutual area, is then defined for a set of n and a set of m points with the same centroid. Using the mutual invariants of two ellipses with a common centre under rotations of the coordinate axes, we obtain some inequalities connecting two sets of points with a common centroid.

The note on this subject published in the Gazette of July 1941 was written in the hope that it might clear away a difficulty which I have good reason to know many students and teachers of dynamics find, when confronted by a problem of changing mass, where it is not obvious which, if indeed either, of the equations M dV/dt = F or d (MV)/dt = F applies. Among letters previously received on the subject I have found it asserted that the latter equation is “fundamental” and that results not in conformity with it must be wrong. I showed by three simple examples of changing mass that it is possible for one or other of these equations to be true or for neither to be true, and hoped that readers would feel satisfied to form the equation of motion for any such case directly, by the simple process of equating the change of momentum in time δt to the impulse of the force producing the change. Could there be a simpler procedure? Unfortunately readers in general are not so easily satisfied, and I seem to have stimulated an ambition to produce an equation of motion “of quite general validity” applicable to all such cases.

Writers on algebra have noticed that the usual laws of indices lead to self contradictions (Chrystal called them “paradoxes”), no matter whether the meaning of a fractional index is defined by

or by

but it does not appear that anyone has yet shown how the laws can be modified in a way that makes them self-consistent. In this note, the necessary amendments are stated for each of these definitions, and are shown to be very much simpler for definition (1). This is fortunate; for that definition appears to be the generally accepted one, though definition (2) is adopted by Hardy and some others.