The purpose of this article is to outline a system of notation for dealing with some of the properties of the general triangle. By means of it results obtained may be concisely stated and, by symbolic manipulation, other results implied but not explicitly realised, may be found.

Suppose C is a convex curve bounding a plane domain of area A. l will denote the length of C and d its diameter ; that is, the upper bound of the distance between any pair of its points. It is clear that, for all convex curves C, 2d≤l≤πd. In this paper I consider classes of convex curves C with a given value of l satisfying one or more further conditions. The aim will be in each case to find the curves which enclose minimum or maximum area. The existence of a curve of a given class ∑ for which the area enclosed attains the maximum or minimum possible follows from compactness in the class of convex curves in a bounded part of the plane.

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.