1. Let α, β, γ, δ be parameters of four points A, B, C, D on the in-circle x2 + y2 = a2, the coordinates of A being (a cos α, a sin α) and so on. (Fig. 1).

The tangent at A is x cos α + y sin α - a = 0. The equation

It is well known that any map on a torus can be coloured with seven colours so that no two contiguous regions are of the same colour, and that fewer than seven colours will not suffice as there exist maps of seven regions of which each one abuts on to each other. In this paper all such maps of seven regions are analysed and a simple method of constructing them is given.

1. The object of this paper may be best explained by reference to the figure. ABC is a triangle with circumcentre O and orthocentre P. What other triangles share these points with ABC?

There is clearly a doubly infinite family of such triangles. If we confine attention for the moment to those possessing the same circumradius R, we shall have a singly infinite family, which we may call the family R. These also possess in common, in addition to circumcentre and orthocentre,

(i) nine-points centre U (the middle point of OP) and nine-points circle, radius ½R ;

(ii) centroid G, where (OGUP) is harmonic.

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.

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.

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

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.