A number of particles are constrained to move freely on the surface of a sphere. Between every pair of them there acts a force which depends only on the distance between them, the law of force being the same for all particles. For a given law of force and a given number of particles, there are a number of configurations of the particles in which they are in equilibrium. In general, these configurations will depend on the law of force: we do not expect the particles to remain in equilibrium if the law of force is changed. But there are certain configurations in which the particles are in equilibrium whatever the law of force may be, for instance if the particles are at the vertices of a regular polyhedron. In this paper these configurations are investigated. It is evident that if a configuration has rotational symmetry about the diameter through a certain particle P of it, i.e. if the configuration can be rotated through a fraction of a revolution about the diameter through P and thereby brought into coincidence with its initial position, then the particle P is in equilibrium whatever the law of force may be. Thus among the configurations with the desired property are those which have this rotational symmetry about every diameter through particles of the system. It is shown that these are in fact the only such configurations. Specifically, the only such configurations are those in which the particles are equally spaced on a great circle, with or without two further particles one at each pole, and those which, in relation to a regular polyhedron, comprise a set of particles at its vertices or a set at the mid-points of its edges or a set at the centres of its faces or any two or all three of these sets taken together. (This classification is redundant but it seems the most convenient.) No discussion of the stability of the configurations is given, as this depends in any given case on the law of force, the present discussion being confined to the property of equilibrium with independence of the law of force. The main proof occupies §§ 2-4. In §5 there is discussed the problem of two or more sets of particles on the sphere which are to be in equilibrium whatever the laws of force may be between particles both of the same set or of different sets. §6 relates the present problem to that of maximising the least distance between any two of an assigned number of particles.

The two essentially different concepts of “form” and “function” are sometimes confused in elementary text-books, and in this note I set out to stress one or two important formal results, in particular the formal interpretation of the binomial theorem for a negative integral exponent. It has been one specific problem, namely the establishing of the Wronski relations

a1-h1=0, a2-a1h1+h2=0, .. ,ak-ak-1h1+...+(-1)khk=0,...

connecting the elementary symmetric functions ar and the homogeneous prodcut sums hr, which has recently focused my attention on this subject and prompted me to write down a few observations.

Consider a regular tetrahedron ABCD of side p (Fig. 1); let P be a point on the face ABC and Q a point on the face BCD; we first consider the geodesic distance between P and Q.

The writer wishes first to acknowledge his indebtedness to Prof. Brodetsky’s book [1] on Nomography; the following observations were, in fact, suggested as a result of reading that work.

Nomography, as most readers will know, is essentially a method of solving problems graphically, but whereas with ordinary graphical solutions a new curve must be drawn each time the data are changed, a nomogram once drawn can be used to solve all problems of a given type (e.g. to solve all equations of the form a cos x +b sin x = l, or to solve all possible pairs of simultaneous linear equations in two unknowns).

The polar equation of the parabola y2=8ax, referred to the origin of coordinates as pole is

r sin2θ = 8a cos θ

or

Inverting with the pole as centre of inversion and 4a as the radius of inversion we find that the equation to the inverse curve is

or r cos θ = 2a sin2θ

or r = 2a sin θ tan θ.

The following error appears in a number of text-books, and tender minds are thereby led astray.

A body (such as a bicycle or railway carriage is travelling in a horizontal circle with uniform speed; G is the centre of gravity of the body, O is the centre of the circle by G, GX is a horizontal line perpendicular to OG and GY is a vertical line.)

Problem. A net is made of string round a unit sphere, so that the sphere cannot slip out of it. How long need the string be?

This problem was offered by me as a “Contest Problem” to undergraduates of Cambridge last Lent Term and a correct solution was given by Mr. R. Schwarzenberger of Trinity College. I am publishing my own solution, as his solution is considerably more complicated.

The traditional two-valued logical calculus is concerned with propositions and their truth-values, true or false. Since the work of George Boole (1) these truth-values have been associated with the symbols 1, 0 respectively. If p denotes a proposition we use the same symbol p to denote its truth-value; thus if p is true we write p = 1 and if p is false we write p = 0. Then derived propositions such as “not p”, “p and q” have them selves truth-values which depend on those of p, q and are thus functions of the truth-values p, q.