Scientific Serials

Abstract

American Journal of Mathematics, vol. xiii., Nos. 2 and 3.—In No. 2 is concluded Part I. of a lengthy article by O. Bolga on the theory of substitution-groups and its application to algebraical equations; the final section discusses groups of operations, especially those obtained from the “groups” of rotations of a regular polyhedron which leave it congruent with its first position. Part II. deals with Galois' theory of algebraic equations.”—The following papers also appear:— “Quelques propriétés des nombres KPm,” by M. M. d'Ocagne. These numbers have been discussed in a previous article (1887, p. 353), where they were defined by means of a triangle analogous to Pascal's;—“Sur les lois de forces centrales faisant décrire à leur point d'application une conique quelles que soient les conditions initiates,” by P. Appell.—On certain identities in the theory of matrices, by H. Taber.— Systems of rays normal to a surface, by W. C. L. Gorton.—On the epicycloid, by F. Morley. Some interesting results of “Wolstenholme's and others are here obtained by the use of circular co-ordinates.—The reduction of by the substitution, by H. P. Manning. A table of available forms is added, and attention drawn to those forms in it given by Cayley (“Elliptic Functions,” p. 316).—A simple statement of proof of reciprocal theorem, by J. C. Field.—Related expressions for Bernouilli's and Euler's numbers, by J. C. Field.—In No. 2 appears a third memoir, on a new theory of symmetric functions, by Major P. A. MacMahon, R.A. Attention is drawn to a fundamental theorem in operations, given without proof. It is a generalization of a theorem by Sylvester which is itself a generalization of Taylor's theorem; “it enables us from any linear function P of the operators to determine another linear function Q, such that exp. P = exp. Q,” the bar in exp. u being used by the author to indicate that the multiplication of operators that occur in u is symbolic.—M. Joseph Perrott also contributes a paper entitled “Remarque au sujet du théoréme d'Euclide sur l'infinité du nombre des nombres premiers.”