We define a Watt quadrilateral to be a quadrilateral with a pair of opposite sides of equal length. See Figure 1.

2. Linkages

The Watt linkage (Figure 2) has equal-length cranks AD and BC, A and B fixed, and coupler bar CD. It was devised by James Watt about 1784 to constrain the steam-engine piston rod, connected at E, the midpoint of CD, to approximate straight-line motion over a limited range.

The “curious” property noted by Mr. A. P. Rollett (Note 1261 in the Mathematical Gazette for December 1937), with regard to the centres of circles touching one another and inscribed in the space between two given circles, leads to an interesting general theorem on all such sets of circles.

Given any two circles, and a set of circles touching one another and the two given circles, then, if r1r2, r3 are the radii of three successive circles of the set, of which r2 is known, r1 and r3 are the roots of a quadratic equation of which the coefficients are simple functions of r2 and the radii of the given circles.

After constructing Platonic solids, Archimedean solids and some of the Kepler-Poinsot polyhedra, I became interested in zonohedra, like the rhombic dodecahedron and triacontahedron shown in Fig. 1. These

models were new to me and most of what I knew about them at that time is summed up by H. S. M. Coxeter in his book Regular polytopes as follows:

Triangles with a 60° angle possess many interesting geometric properties. Here is a well-known example: In ΔABC with incentre I, let the rays meet the sides AC, AB at B1, C1 respectively. If ∠B = ∠C, then IB1 = IC1, trivially. Does the converse implication hold? No – the same condition holds when ∠A = 60° (irrespective of the relation between ∠B and ∠C). In this article we study some more properties possessed by triangles having a 60° angle.

Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.

It is well known that the six rotations of an equilateral triangle form a group; that is to say that, if an equilateral triangle ABC occupies a certain triangular space, there are six operations and six only which when performed either singly or in succession will leave the triangle still occupying the same space, but with vertices in different positions. The set of six possible orientations of the movable triangle are drawn below and numbered.

In this article an attempt has been made to indicate the ordinary methods of integration of the total differential equation

Then the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is