The death of the greatest English mathematician of our time is no mere national loss, for Hardy was recognised throughout the mathematical world as a master of our science. Of his studies, Hardy himself said, in his Inaugural Lecture at Oxford: “What we do may be small, but it has a certain character of permanence”. Few of us would dare to call Hardy’s contribution to mathematics small, while none of us can have any doubt about the lasting nature of his work. For a full account of the new pathways he opened up, the new territories he explored, reference must be made to the notices being prepared for the Royal Society and the London Mathematical Society. In the Gazette, it is fitting that we should record with special emphasis the debt which teachers of mathematics in this country owe to Hardy for the vast improvements in the teaching of analysis during the past forty years, from vagueness to precision, from obscurity to clarity, along lines mapped out and laid down for us by him.

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.

When we want to use algebraic methods to discuss metrical properties of the triangle such as Feuerbach’s theorem, Simson’s line, etc., we have the dilemma that Cartesian coordinates confuse the symmetry of the triangle, whereas in areals or other triangular coordinates the metrical formulae are not sufficiently simple. A convenient alternative is to use the method of the Argand diagram: that is, to use complex coordinates (or “affixes”) x ± iy.

The equation to be solved is

where a, b, ƒ are real parameters. (1) is non-linear by virtue of the term by3. Suppose that the free end of a restoring device or spring is fixed to a mass m resting on a frictionless horizontal plane. Let the force-displacement relationship of the spring ƒ1 = g(y), a function of the displacement y. If the mass is driven by an external force F cos ωt, the equation of motion is

or

In (l), g(y/m = ay + by3 and ƒ = F/m. If b = 0, the spring stiffness is ma, a constant, so (3) is linear. When b ≠ 0, the stiffness is not constant, but has the value mdg(y)/dy = m(a + 3by2). This increases or decreases with increase in y, according as b ≷ 0, and (1) is then non-linear. Putting a = (ω2 + λ2), (1) becomes

This equation may be considered to symbolise a simple (linear) mass-spring system of free pulsatance ω, driven by a force ƒ cos ωt – (λ2y + by3).

I wonder why problems about map-colourings are so fascinating? I know several people who have made more or less serious attempts to prove the Four-Colour Theorem, and I suppose many more have made collections of maps in the hope of hitting upon a counter-example. I like P. G. Tait’s approach myself; he removed the problem from the plane so that it could be discussed in terms of more general figures. He showed that the Four-Colour Theorem is equivalent to the proposition that if N is a connected cubical network, without an isthmus, in the plane, then the edges of N can be coloured in three colours so that the colours of the three meeting at any vertex are all different. (A cubical network is a set of points called vertices joined in pairs by simple arcs called edges, no two of which intersect except at a common vertex, in such a way that just three edges meet at each vertex. An isthmus is an edge whose removal destroys the connection of the network.) It was at first conjectured that every cubical network having no isthmus could be “three-coloured” in this way, but this was disproved by the example of Fig. 1, for which it may readily be verified that no three-colouring exists.

In certain problems of X-ray crystallography it became necessary to determine the coordinates of a point P, originally at (x, y, z) in the non-rectangular, Cartesian system (X, Y, Z) after rotation of the vector about some line . This is the classical Euler transformation, and the result is well known for rectangular axes. Since, however, no reference could be found to the more general case, the results of this calculation may be of interest.

If we take any integral number, and suppose it written round the circumference of a circle, it is possible to form differences by subtracting each digit from its neighbour on the right. If the result is negative, 10 is added to bring the answer positive. The second and successive differences may be formed similarly. It is found that at some stage the lines of differences start to repeat those occurring before.