So, for example,

x appearing 3 times,

and x appearing 5 times.

The hyperexponential function is then defined as

for those values of x for which this limit exists. Since an infinite number of operations are involved, we clearly need to study carefully the question of convergence.

During an investigation on some Diophantine systems, we were led to consider the following set

where x, y, a, b, c are indeterminate integers.

From elementary algebra we know that, if they exist, x and y are the roots of the quadratic equation

In classical dynamics, Newton’s laws hold in all inertial frames. In particular, the work done by a force along the path traversed by a particle equals the change in kinetic energy of the particle. However, velocities and paths are frame dependent. Thus we investigate the work of a force in two inertial frames. We see that work is frame dependent. Furthermore, we determine the relation between work in two inertial frames.

In the second named author presented some results and problems concerning interlinked loops. These were ‘inspired’ by the well-known Borromean rings – three loops with the property that they are interlinked, but if we cut and remove any one of the loops, then the remaining two loops are not linked and so can be separated.

To conform more closely with standard terminology in this area we shall here use the term ‘link of n components’ to denote such a set of loops (rather than linkage). In all the cases we consider, the individual loops that make up each link are unknotted.

A geoboard is a flat piece of wood with nails driven in at integer lattice points, and polygons are constructed on the geoboard by placing rubber bands over the nails. At a workshop for teachers, held at Stellenbosch in South Africa, and organised by the African Institute for Mathematical Sciences Schools Enrichment Centre (see http://aims.ac.za/aimssec), it was conjectured that if two polygons on a geoboard overlap, then the the area of the overlap is rational. It is not difficult to prove this but, as we shall see, the ideas in the proof go far beyond the original problem and may be of interest to some Gazette readers.

The concept of isogonal conjugate points and isotomic conjugate points relative to a given triangle appears to have been introduced by Casey in 1888. Since then these topics have been treated in many important books, such as Johnson Coxeter and Greitzer and more recently Honsberger. The idea originated from the fact that the isogonal conjugates of the points on a fixed line form a circumconic of the triangle involved. That certain pairs of points are isogonal conjugates is of considerable theoretical significance. For example, that the symmedian point K in a triangle is the isogonal conjugate of its centroid G is an important result in understanding the significance of the symmedian point. The concepts involved, though well known to enthusiasts of triangle geometry, deserve additional publicity.

In Euler derives the approximate summation formula

This primitive result does not appear to be well known to modern readers, and we will try to show by examples that it deserves more attention. After translating this paper into English and reflecting on its contents, we thought it desirable to present Euler’s work in a form convenient for today’s readers. Because of inadequate notations for functions, sums, and definite integrals, the modern reader is likely to find Euler’s writing difficult, even in translation. Also, in Euler’s time, mathematicians were familiar with properties of the conic sections that have largely been ignored today.

A ballistic missile trajectory consists of at least three sectors:

• 1. Powered flight from launch until engine cut-off, the point we shall refer to as the burnout point.

• 2. Free flight, which covers most of the trajectory.

• 3. Re-entry, which begins at some point where atmospheric drag forces become a significant factor in determining the subsequent motion of the missile, this sector lasting until impact with the surface of the earth which we shall refer to generally as the ground.

During the free flight sector, the trajectory is part of a conic section—invariably a sector of an elliptic orbit which intercepts the ground—the subject of this article. Figure 1 illustrates some of the important parameters in the analysis. We are concerned here with the free flight sector of the trajectory, which can be treated without any knowledge of the details of the powered flight phase.