The following problem was included in the paper set in the Mathematical Tripos on the morning of January 5, 1881 :

iii. Prove that, if a + b + c = 0 and x + y + z = 0, then 4(ax + by + cz)3 - 3(ax+ by + cz)( a2+ b2+ c2)(a2+ y2 + z2) - 2 (b - c) (c - a) (a - b)(y - z) (z - x) (z - y)=54abcxyz.

A solution of this problem was published by A. Cayley in June 1881 ; see the Messenger of Mathematics, XI. (1882), 23-25. Cayley's solution took the form of a long and cumbrous verification, and he admitted that he did not know the origin of the result nor did he see any simple way of proving it.

The centre of mass, G, of a system of particles (masses mr at Pr) can be defined by the relation, O being an arbitrary point,

This is equivalent to finding the point Q1 which divides P1P2 in the ratio m2 : m1 ; then the point Q2 which divides Q1P3 in the ratio m3 : (m1 + m2) ; then the point Q3 which divides Q2P4 in the ratio m4 : (m1 + m2 + m3), and so on until finally G is reached.

A paradox connected with the idea of a uniform field of force is recalled. The main purpose is to discuss frames of reference in the gravitational fields of systems originally studied by Milne and McCrea in developing their “newtonian cosmology” It is shown that a frame moving with any particle of the material of such a system serves as a newtonian frame for the description of mechanical phenomena throughout a region that may be taken to be arbitrarily large and that a frame of such a sort is the nearest approach possible in the system to the realization of a local newtonian frame ; any two such frames are nevertheless in mutually accelerated motion.

1. The problem of forming a necklace in which no two adjacent beads have the same colour was found, unexpectedly, to lead to the little Fermat theorem and to a companion congruence with a composite modulus.

Let there be N + 1 different bead-colours. Then there are N + 1 choices of colour for the first bead to go on the string, and for each bead thereafter there are N choices. Hence for n beads the number of arrangements is (N + 1) Nn-1 But if the nth bead is the last to go on, it must have a different colour from the first.

I have never seen any explanation of why it is possible for a fast bowler to make a new cricket ball swerve, and why it becomes more difficult or impossible to do this when the shine has worn off the ball. Of course a sliced golf ball swerves because of the spin it has, and this swerve is a well-known illustration of the Magnus effect. Spin cannot be the explanation in the case of fast swerve bowling, because if it were the effect would increase as the ball became older and rougher. In any case a fast bowler imparts little or no spin to the ball.

The purpose of this note is to suggest a system of marking chess-players that is both simple to apply and based on a theoretical background. The non-mathematical reader will not need to understand the formulae, but is recommended to read all the words.

In the British Chess Federation Year Book, 1953–4, pages 39–42, there is an article entitled “ A grading system for British players ”. It describes a system that it admits to be open to some objections. At first sight the system looks reasonable and is fairly simple to apply, though of course any system requires a fair amount of calculation if it is to be applied to hundreds of players. An objection to the system, mentioned in the article, is that it is always profitable to play an opponent whose mark exceeds one's own by more than 500 points, unless the system is modified in an inelegant manner. The publication of the article is stimulating to further thought and the system to be described here is one of the results. It is not open to an objection of the above form.

The following paragraphs have been assembled in consequence of my reading Dr. Cundy’s note on 25-point geometry. Towards the end of it, apparently mindful of the adjunction of a “ line at infinity ” to the Euclidean plane, he adjoins a line to the 25-point plane and so obtains a geometry of 31 points. Here I reverse this procedure : I start with the 31-point geometry and thereafter assign to one of its 31 lines the rôle of the “ line at infinity ”. This seems more in the spirit of Cayley's dictum at the end of his Sixth Memoir on Quantics, that “ descriptive geometry is all geometry ” and metrical geometry only a part thereof.

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.

Dr. S. Vajda commences Note 2159 (Math. Gazette, Vol. 34, p. 211) with the following :

In his review of H. S. M. Coxeters Regular Polytopes (Math. Gazette Vol. 33, p. 49) Mr. H. Martyn Cundy quotes the formula

and supplies the proof by considering the coefficients of xn in the expansion of

In cases where successive integration of F(x) is not practical, a solution of the equation d2y/dx2=F(x) is obtained by approximation. The solution attempted is to find the value of y at selected points in the range of integration. A common procedure is to reduce the problem to the solution of a set of linear simultaneous equations, and these equations are then solved by “relaxation”. It does not appear to be generally known that an explicit and simple solution of these simultaneous equations exists. In this note the detail of such a solution is carried out for functions F(x) as occur in the theory of thin beams.

The longitudinal oscillation of “light” springs carrying loads, in various combinations, has a well-established place in all mechanics textbooks, but none of the standard works appear to deal thoroughly with the oscillation of a spring whose mass is not negligible. The purpose of this note is to investigate the vertical longitudinal oscillation of a heavy spring, first by itself and secondly when carrying a load. It will be seen that in each case the motion consists not of a single S.H.M. (as for the light spring), but an infinite number of S.H.M.s ; when the spring oscillates by itself the frequency of these S.H.M.s are multiples of a fundamental frequency, but when the spring carries a load the frequencies are not so simply related.

In the Gazette for December 1953, I gave an account of all that I knew about Schur’s inequality, namely

If μ≥0 and x, y, z are all positive, then

with equality occurring when and only whenx = y = z.