§1. The following investigation arose out of a reconsideration of a certain invariant of six points in space

I = (1234)(3456)(5612) – (2345)(4561)(6123)

where (1234) denotes the determinant whose rows are the coordinates of the first four points: and similarly for the other factors. The vanishing of this invariant implies that the sixth point lies upon a certain quadric defined by the other five. The following elementary account brings out several interesting and possibly new theorems upon quadric surfaces.

In connection with some work on the theory of probable errors, the following series identity arose:—

where a is a positive integer.

As the main interest of this identity appears to lie in itself and in the method of proof, and not in its connection with the theory of probable errors, it seems worth while to record it separately.

1. Between the solutions of the equations

a relation can be established, provided the functional symbols f, ø are inverse to one another. For example, let f (x) = sin x, then ø(ξ)= arcsin ξ, and the two equations are

The process consists in obtaining from (1) the second order equation satisfied by yx = dy/dx, making a change of independent variable in (2), and comparing the resulting equations.

§ 1. Chowla has generalised Wolstenholme's Theorem as follows:

where p denotes as usual a prime, and < is used for “less than and prime to.”

In order that an algebraic series of a finite number of terms may be summed in simple form, there must exist a difference equation satisfied by the terms. But, owing to the fact that no general method of finding this equation is given in textbooks, the beginner does not acquire confidence in summation until his experience is wide enough to include all of the usual types of such series; and even then he may come to the conclusion that the finding of the difference equation is no more than a lucky chance. It is therefore proposed in the following notes to show that the ordinary types of summable terminating series can be reduced to one, and to find a single general expression for the sum. It will also be shown that the criterion of summability is the “convergency ratio.”

By defining a logarithm as

we may visualise the function as the area under the curve

, measured to the right from the zero value at the ordinate AB. The fundamental properties follow at once from (1): for if u = av, then

The theorem that a line cutting a pair of conies in four harmonically separated points envelopes a conic, called the Φ conic, is a familiar result which admits of a simple proof by analytical methods. A synthetic proof, however, if we exclude the use of (2, 2) correspondences, is rather elusive. I have not been able to find such a proof in any book, and the only one published as far as I am aware is that set as a question in the 1934 Mathematical Tripos, due to Mr F. P. White. The proof written out below is rather more direct and may therefore be worth recording.

The expansion, in the form of a series, of the quotient of two determinants, in which the numerator determinant differs from the denominator in one column only, is moderately well known. It was found in 1825 by P. Schweins (see Muir's History, Vol. 1, pp. 171–2), and has many times since been rediscovered or proved ad hoc for particular applications.