In view of the difficulty of arranging for meetings under existing conditions, the Council has set up an Executive Committee to conduct the business usually transacted by the Council. This Committee consists of the President, the Treasurer, the Secretaries, the Librarian, the Editor of the Mathematical Gazette, the Chairman and the Secretary of the Teaching Committee, together with Mr. Siddons, Mr. Hope-Jones and Miss Punnett.

My remarks this morning will be devoted mainly to general considerations and to principles. Dr. Topping and Mr. Verity are to deal with the courses in detail, but behind all that they say there must be a background of principle, and I hope that a statement of my creed will not find them active dissenters.

I take as my text a quotation from Ruskin which I placed on the flyleaf of a little book I wrote in 1925, and of the truth of which I am as convinced as ever—“Know what you have to do, and do it, for I believe that failure is less frequently attributable to either insufficiency of means or impatience of labour, than to a confused understanding of the thing actually to be done.”

In a previous paper, which appeared in The Mathematical Gazette for December 1938, the writer showed that, assuming the concept of a rigid rod, it is possible to derive the fundamental formulae of special Relativity from the consideration of photographs taken by two observers in relative uniform motion.

The extension of this method of approach to the consideration of the relativity of time measures appears to furnish some interesting results.

It is economical in the use of symbols to express the sides of a triangle fractionally, by dividing the expressions for a, b, c given in my previous Note (Gazette, XXIII, October, 1939, Note 1398, p. 380) by qy. Thus writing h for x/y and K for p/q, we have

a = hk + h − k + 1,

b = hk − h + k + 1,

c = 2(h + k),

and n = hk − 1,

where l, m, n are the medians bisecting a, b, c.

In two recent Notes (1354 and 1289) formulae have been given for the determination of some, but not all, triangles having integral sides and one integral median. All such triangles can be found, with the aid of a list of primes of the form 4n + 1.

From the equation AB 2 + AC 2 = 2AD 2 + 2BD 2 it follows that if AB, AC and AD are to be integral, then 2BD 2 is an integer, whence, if BC is to be integral, BD is an integer.

The theorem of J E. Littlewood stated below includes Maclaurin’s integral test and furnishes the basis for a unified theory of convergence criteria for series of positive terms.

Theorem A. If F(x) is positive monotone decreasing and (Dn) is such that

then converges or diverges with .