The reciprocal nomogram described in Note 2108 (February, 1950) for the solution of 1/R=1/R1 + 1/R2 is a particular case of a more general property.

The classical propositions of the elementary theory of numbers concerning divisibility, primes and linear indeterminate equations, are all based on the possibility of division with a remainder.

This note gives a general 2nd order solution, with the corresponding 3rd order correction, by means of which a good numerical answer can be obtained from an approximate estimate in one operation, and results of high accuracy by one or two iterations. The method is applied to integral roots of numbers, and to common types of equations.

For the extraction of square roots and cube roots of high accuracy, the general method is developed in the form of series designed to simplify the computation, and one example is given in full with an accuracy of over 30 significant figures.

It is a sound principle in teaching that we should begin by saying what a thing is, rather than what it is not, but the reader of the Gazette is not coming across the word involution for the first time, anti the best serrice I can render him is to insist at the outset that involution is not a form of homography Before I can enlarge on this warning, I must add another. In elementary work we are in no danger of confusing the relation of collinearity, a relation between three or more points, with a class of collinear points, for which we have the familiar name of line. But the one word involution has been made to do duty both for a relation and for a class, and although when we understand the subject we can always interpret the word froin the context, the teacher will find it worth while to sacrifice brevity for a time and to tallr of the involution-relation and of an involution-class.

For the last 75 years the largest known prime has been the Mersenne number 2127 – 1, proved prime by Lucas. In two recent notes (Eureka, October 1951 and Nature, 168 (1951), 838), Miller and Wheeler announced their identification of several larger primes, of which the greatest is

Let a1, a2, a3 denote the sides α1, α2, α3 the angles and A1, A2, A3 the vertices of a triangle in the Euclidean plane. A point P, whose distances from the sides α1, α2, α3 are in the ratio p1: p2: p3 will be denoted by P[p1]. p1, p2, p3 are called the normal coordinates of P. Thus the unit point E[l] is the incentro of the triangle, S[cosec α1] is the centroid, M[cos α1] is the circumcentre, and H[sec α1] is the orthocentre. Similarly, using normal line coordinates, we have l0 [l] is the unit line, l∞[sin α1] is the line at infinity and lH[cos α1] is the axis of the altitudes.

In the Journal of the London Mathematical Society (July, 1951, Vol. 26, pp. 221-8), proofs are given of a theorem of McCoy's and of various results that follow from it. McCoy's theorem establishes the necessary and sufficient condition that a number of n × n matrices should be reducible to triangular form by the same transformation T-1( )T, where the matrix T is unitary, (We define a triangular matrix to be one in which all the elements below the diagonal are zero.)

Direct and independent proof is given to the theorem, implicit in the possibility, and fundamental for the procedure of the simultaneous canonical reduction of a pair of real quadratic forms, one of which is positive definite. In this way a complete unification of the possibility and the procedure is effected.

Recently Ferrar (1) has considered this question of reduction by a method “free of difficult invariant factor arguments”, and Todd (2) the example of the orthogonal reduction of a single form in a direct and non-inductive fashion to remove, with reference to the traditional method for proving the possibility, “the aesthetic objection that it does not correspond to the practical method for obtaining the reduction”.

The concept of the direct product is widely used in the theory of groups, algebras, vector spaces and allied topics, but its use as a manipulative tool in matrix algebra is less common.

The present note shows how the direct product of matrices arises in elementary work, derives some of its more important properties, and gives some examples of its use.