That every person in the realm who teaches mathematics should be a member of our Association is an ideal towards which we are converging non-uniformly, if at all. Were this ideal state of things a reality no apology would be needed for a paper dealing with topics which though educationally important are from a mathematical point of view very familiar. As it is, the feeling haunted me as I wrote that I was writing a sermon, and that the people whom it might benefit would not be in the room, were not even members of our Association. As, however, other writers of sermons must, I suppose, sometimes find themselves in a similar position, 1 have ventured to go on.

In the first place, the present paper refers to conditions prevailing in England only, to the exclusion of Scotland, Ireland and Wales.

Secondly, the term " public school" has, in England, a rather special meaning. A public school is generally understood to be an endowed secondary school of fair size, educating boys of the upper and middle classes, independent of state control except in so far as its income is supplemented from public funds. Public schools may be either boarding schools or day schools, but most of them belong to the former category.

Binal Fractions. The reciprocals (1/N) of any number (N) will be termed Binal Fractions, or shortly Binals, when expressed in the binary scale. The notation used is similar to that of ordinary decimals. Thus (see table at end):

1°. A point is used to mark the beginning of the fraction.

2°. The binal (1/N) consists of a certain sequence of ciphers and units (the only digits used) to the right of the point; partly non-recurrent and partly recurrent.