The connection between thought and its expression in symbolism is perhaps more important in mathematics than in any other subject, for the complexity of mathematical thinking soon makes reference to symbolism an indispensable method of learning and discovery This is true both in the historical development of the science and in the experience of the individual student. Without visible symbolism to act as a scaffolding, the sheer complexity of the subject matter results very soon in an incoherence and breakdown of the thinking process. While this is true to some extent of all subjects of any high degree of complexity, in mathematics a new and extremely interesting element enters into the situation.

The County Cricket Championship table that appeared on 16th June disclosed a paradoxical state of affairs, which, when examined closely, is shown to depend upon the neglect of an important statistical principle used in the calculation of death-rates, comparative wage-rates, index numbers, and other data not usually associated with our national game. Sussex and Yorkshire had each completed seven matches, and each had won six and lost one. As for the unfinished matches, Sussex had four times led on the first innings and once been behind, while Yorkshire had been behind on the first innings in the only two unfinished matches they played. Thus on the complete matches the two counties were exactly equal, while on the unfinished ones Sussex had done better.

When I was younger this was a very well-known subject, and actually was called Science Subject No. 1 in the days of the Board of Education’s South Kensington Examination. Engineers (mechanical and civil) and craftsmen were taught it as a fundamental subject, and even teachers in training were expected to know something about it. The rise of Practical Mathematics put it into the background—wrongly I think—but there are signs that a healthy interest is being taken in this section again.

N.B.—In what follows we consider the conic as the projection of the circle and assume only its descriptive (including cross-ratio) properties as known.

1. A circle, centre g1 is cut by a secant line at ww′. Any line cpp′ through c, the pole of ww′, cuts the circle at p and again at p′. The lines wp and w′p′ intersect at d. The locus of d for the variable secant line through c is a hyperbola touching the given circle at ww′ (If w′p and wp′ meet at d′, the locus of d′ is the same hyperbola.)

We have shown how the theory of the Fractional Calculus enables us to expand functions in various forms. In the present paper we shall obtain some further expansions.

A λth integral of f(z) along a simple curve l is defined by

where γ is the least integer greater than or equal to zero such that R(λ) + γ>0, D stands for d/dz, and the integration and differentiation are along l.

This paper is a sequel to a paper entitled “The Harmonic Conies”, Math. Gazette, Vol. XIX, page 98. How is it that text books on Algebraic Geometry deal with invariants and covariante of two conies yet never so much as hint at the existence of such forms for more than two conies?