Geometrical Optics as a part of the mathematical curriculum in schools, universities, and technical colleges is now a dead subject, yet the generalised study of ray-tracing in optics started by Sir William Rowan Hamilton in 1824 is well worth attention on all sides. He writes: “Those who have meditated on the beauty and utility in theoretical mechanics of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced in the Mécanique Analytique must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics when it shall possess an appropriate method and become the unfolding of a central idea.” With this end in view, Sir Wm. Hamilton submits his three papers on the Theory of Systems of Rays to the Royal Irish Academy, which, as Lord Rayleigh said, bear the mark of genius throughout. The enormous task that Sir Wm. Hamilton set himself will be evident when we discuss the wanderings of rays through optical systems in an elementary way to-night leading up to Hamilton’s general method of the Characteristic Function and its development.

This is a clumsily long title, but it might have been much more lengthy if it had gone on to suggest how the problem in probability arose. That would involve us in many questions, so we may say at once that the problem is equivalent to the following: If from a bag containing M different balls N drawings are made, where M, N are very large numbers (the ball being replaced after each drawing), what is approximately the chance that exactly r balls should fail to appear?

Every attempt to give an elementary rigorous course of Analysis is beset by several unsurmountable difficulties. Every teacher of the subject will readily agree that the two main obstacles are (i) Dedekind’s theorem, and (ii) the Heine-Borel theorem. Of these the first may be rendered less apparent, for in an elementary treatment there is no great objection to the postulation of the existence of an upper and a lower bound for every bounded linear set of numbers. The second cannot, however, be avoided, for the important theorems of the differential calculus, such as Rolle’s theorem, the Mean-value theorem, Cauchy’s formula, and Taylor’s theorem can only be rigorously proved by appealing to certain properties of continuous functions, which in their turn depend upon the Heine-Borel theorem, or upon some alternative simpler theorem on intervals such as the one which Carslaw uses to replace it.