I Feel it a great honour to be permitted to address you to-day There have been some schoolmaster Presidents in the history of the Association. R. B. Hayward was President from 1878 to 1888 while a master at Harrow; Canon J. M. Wilson, a pioneer of Science teaching at Rugby, and one of the earliest to work actively for geometrical reform, became our President long after his retirement, and then there have been three in recent times, with all of whom I have been closely associated, and to whom I should wish to pay tribute.

Introduction. If we call the number pa + qb the Tp,q transform of the number n = 10a + b, then in general the Tp,q transform of n will differ from n; it easily seen, however, that given any number n, we can always find a pair p, such that the Tp.q transform of n is n itself. In fact, if n = 10a + b and p = 10 − b q = a + 1 then the Tp,q transform of n is (10 − b) a + (a + 1)b = 10a + b = n.

It is the intention of this paper broadly to survey present possibilities of work for honours graduates in mathematics. Detailed information, which obscures the picture and soon becomes out of date, will not be given.