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All points in an n-space equidistant from a fixed point (the centre) constitute what may be called a spherical continuum of the nth order,—the continuum being of n — 1 dimensions ((n — l)-dimensional spread) and of the 2nd degree. Any region of this spherical continuum bounded by n (n — l)-dimensional linear continua or, primes (spaces of n — 1 dimensions), passing through the centre shall be called a spherical simplex of the nth order. This spherical simplex is bounded by n faces, spherical simplexes of the (n — l)th order, each of which in turn is bounded by n — 1 spherical simplexes of the (n — 2)th order, and so on till we reach spherical triangles, arcs and lastly points, the vertices. The total number of spherical simplexes of different orders connected with one of the nth order is 2n — 2. The n spherical continua of the (n — l)th order which contain the faces of the spherical simplex of the nth order determine a set of 2n spherical simplexes of the same order, 2n–1 pairs, the two spherical simplexes of a pair being symmetrically situated with respect to the centre and therefore congruent.
These notes are intended to be read in connexion with Dr A. C. Aitken's paper, Proc. Edinburgh Math. Soc. (2) 1 (1929), 199-203. It is proposed to show (by a simple line of direct algebraic demonstration which is also applicable to the original formula) that Aitken's Theorem can be extended to the Everett types, i.e. the types which include two sets of terms—one set involving u (0) and the resultant of generalised operations on u (0), and the other set involving u (1) and the resultant of similar operations on u (1).
The various formulae for the Legendre Functions, and the relations between these formulae, have been studied by Kummer,Riemann, Olbricht, Hobson, Barnes, Whipple, and others. Hobson obtained some of the relations directly, by expres ing the functions as Pochhammer integrals, and expanding in a number of series each with its own region of convergence. To obtain some of the other formulae, such as (i) below, he transformed the differential equation, and then expressed the functions in terms of the solutions of the transformed equation. Barnes succeeded, by means of his wellknown integrals involving Gamma Functions, in deducing all the formulae directly from the formulae which define the functions.Notes on the history of the subject and references to previous work will be found in the papers by Hobson and Barnes.
The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated that
where s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.
It was proved by Salmon (Geom. of three dimensions (1882), p. 331) that the chords of the curve of intersection of two algebraic surfaces of order m and n. which can be drawn from an arbitrary point,meet the curve upon a surface of order (m — 1) (n — 1); it was proved by Valentiner (Acta Math. 2 (1883), p. 191), and by Noether (Berlin. Abh. (1882), Zur Grundlegung u.s.w., p. 27), that the surface of order (m — 1) (n — 1) is a cone, with vertex at the point from which the chords are drawn; and a converse theorem was given by Halphen (J. de l' école Polyt. 52 (1882), p. 106). But the proofs given by Valentiner and Noether have not the elementary character that seems desirable, Noether's proof in particular depending on the theory of the canonical series upon the curve.