Scientific Serials

Abstract

American Journal of Mathematics, vol. ix. No. 2, January.—The number opens with a continuation of Mr. Greenhill's memoir, wave-motion in hydrodynamics, in which is discussed wave-motion in the following cases: § 21, across a channel with sides sloping at any angle; § 22, against a uniformly-sloping shore; § 24, in a cone; § 25, in a cylinder; and § 23 contains an algebraical solution of waves against a shore.—Prof. Sylvester's lectures on the theory of reciprocants give notes of lectures xvii. to xxiv., with an extract from a letter of M. Halphen in which the existence of invariants in general is established a priori; this is given as introductory to the theory of differential invariants.—A memoir in the theory of numbers, by A. S. Hathaway, contains an historical introduction of interest. The second part considers fundamental principles and definitions, then a problem and the consequences of its solution, and then turns the question of ideal solution of the problem into the question of the establishment of a given theory of ideals; the demonstrations are left for the reader to supply. The third part is occupied with a rigorous establishment of the theory of ideals indicated in the second part.—The next paper, on a theorem respecting the singularities of curves of multiple curvature, by H. B. Fine, is a generalisation of a portion of a previous paper (vol. viii. No. 2) by the same writer.—The number closes with two short notes—one on pencils of conies, by H. D. Thompson (let the eight points in which a conic intersects a quartic be divided into two groups of four, and a conic be passed through each group: the two residual—four-point—groups lie on a conic; an exceptional case in Cayley's theorem, which had been overlooked by the author, is mentioned and references given to where it is discussed); the other consists of observations on the generating functions of the theory of invariants, by Capt. P. A. Macmahon.