1. A method of characterising plane curves was given by Plücker, who showed that the degree (n), the class (m), and the numbers of nodes (δ), bitangents (τ), cusps (κ), and inflexional tangents (ι) are connected by three independent relations. He showed also that n and m, δ and τ, κ and ι are dual pairs.

A classical theorem of Cantor states that the class of all sub-classes of a given class has a cardinal greater than that of the given class. This theorem is here established in a sharpened form, which was suggested to me by a question set by Professor J. M. Whittaker, F.R.S, in the 1950 examination for the Honours B.Sc. Degree at Liverpool.

More than 300 years ago Mersenne made pronouncements as to the prime or composite nature of 2P – 1 for all prime values of p from 1 to 257. His reasons were not disclosed, but the combined efforts of many mathematicians have shown that his statements were substantially correct. Of course when p is composite, two or more factors of 2p ± 1 will often be obvious but, as far as I am aware, only two general theorems relating to a non-obvious factor have been proved. I give these by way of introduction to this article and thereafter proceed to enunciate seven new and original theorems. As I offer no theoretical proofs of the theorems, they must be regarded as conjectures. They are, however, based on extensive computation which has engaged me for a period of three years.

Quadratic Equations. The device is designed to give the numerical value of the numerically smaller root of the equation x2 – px + q = 0, the other root and the signs of the roots α, β being found thereafter from the relations α + β = p, α β = q.

Let A and B be square matrices of the same order, with elements in any field F; it is well known that the characteristic polynomials of AB and BA are the same (see, e.g. C. C. Macduffee, Theory of Matrices, p. 23). The proof of this is easy when one at least of the matrices is non-singular; the object of the following remarks (which are not claimed as original) is to point out that the case | A | = | B | = 0 is just as easy. If one attempts to deduce the result in this case from the result in the non-singular case, unnecessary restrictions on the field F are apt to appear (see e.g., W. V. Parker, American Mathematical Monthly, vol. 60 (1953) p. 316). If one proceeds directly to the general case, no difficulties are encountered.

1. Introduction. For a many-valued function f(z) of the complex variable z, a Riemann surface can be constructed such that, at any point z on the surface, the function has only one value; a function normally multiform, is therefore uniform on a certain Riemann surface.