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    American Journal of Mathematics, vol. xii. No. 1, and index to vols. i.–x. (Baltimore, 1889).—This volume opens with an instalment of sixty pages of a memoir by A. R. Forsyth, F.R.S., on “Systems of Ternariants that are Algebraically Complete.” In this the writer has found it convenient to use “‘ternariant’ as a generic term for concomitants of ternary quantics, instead of giving it the signification which Prof. Sylvester proposed (Amer. J. of Math., vol. v. p. 81) to give to it, viz. the leading coefficients of those concomitants.” The memoir is divided into three parts, and deals with the theory of the algebraically independent concomitants of ternary quantics, taking as the starting-point the six linear partial differential equations of the first order satisfied by them. References are supplied to numerous memoirs on the subject.—Captain (now Major) P. A. Macmahon continues (pp. 61—102) his investigations (vol. xi. No. 1) in a “Second Memoir on a New Theory of Symmetric Functions.” Herein he is engaged with functions which are not necessarily integral, but require partitions, with positive, zero, and negative parts for their symbolical expression. The author thus summarizes his results: (1) a simple proof of a generalized Vandermonde-Waring power law which presents itself in the guise of an invariantive property of a transcendental transformation; (2) the law of “groups of separations”; (3) the fundamental law of algebraic reciprocity; (4) the fundamental law of algebraic expressibility which asserts that certain indicated symmetric functions can be exhibited as linear functions of the separations of any given partition; (5) the existence is established of a pair of symmetrical tables in association with every partition into positive, zero, and negative parts, of every number, positive, zero, or negative.—The closing portion of the number (pp. 103–114) is taken up with an article entitled “De l'Homographie en Mécanique,” by P. Appell.—A likeness of M. Poincaré faces p. 1.—The index is of a twofold description—of authors and of subjects. From the forewords we learn that papers have been published from eighty-nine contributors; these comprise “most of the leading mathematicians of the world.”

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    Scientific Serials. Nature 41, 71 (1889). https://doi.org/10.1038/041071a0

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