All the commonly used rules for the approximate quadrature of areas, such as those of Cotes, Simpson, Tchebychef and Gauss, are based on the assumption that y can be expressed as a rational integral function of x with finite coefficients. A tacit assumption is thus made that is not infinite within the range considered, and it is therefore hardly a matter for surprise that the degree of accuracy obtainable by the use of these rules in the case of a curve which touches the end ordinates is very poor.

We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:

Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.

This paper shows briefly how, for a singly infinite family of curves on a given surface, the fundamental properties of the family at any point are associated with three central conies determined by the curves, in a manner resembling that in which the curvature properties of a surface at any point are associated with Dupin's indicatrix. The differential invariants employed are the two-parametric invariants for the given surface.

The matrices considered in the following note are non-singular, and are related to a given matrix A having elements taken from the field of positive and negative integers and zero. The invariant properties of such a matrix A, under multiplications by matrices of determinant equal to unity, can be formulated, as is well known, in terms of the “elementary divisors” of the determinant |A|. Thus if A is of the nth order, and p is a prime occurring in |A| to the power hn, in the H.C.F. of the first minors of |A| to the power hn−1, in the H.C.F. of the second minors to the power hn−2, and so on, h0 by convention being zero, then the first differences of the h's,

are invariant under the transformations considered, and it is known that r ≥ er−1. The numbers

where the product includes all prime factors of |A|, are called the elementary divisors of |A| and

The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that

In connection with Bertrand's algebraical exercise of 1850, Muir remarks that it is not unlikely that the divisibility of a3 + b3 + c3 − 3abc by a + b + c had been previously noted, although. there is no record of the fact. Bertrand's exercise is to the effect that the circulant of the third order repeats under multiplication or, what is the same, admits of composition; the formulae of composition are stated in the exercise precisely as they would follow from Spottiswoode's theorem on the linear factors of a circulant. The whole of this is implied as an immediate special case, and indeed as one that any reader would construct at once, in an identity due to Lagrange, reproduced by Legendre.

Let the Nodal cubic have the equation

or,

Three points t1, t2, t3 form an apolar triad to the curve when they satisfy an equation of the form