Scientific Serials

Abstract

Bulletin of the American Mathematical Society (January).— Report on the theory of projective invariants: the chief contributions of a decade (by Prof. H. S. White), was read before Section A of the American Association for the Advance, ment of Science in August 1898. The starting-point is from the publication of the second part of Gordan's “Vorlesungen über Invariantentheorie “in 1887. Gordan's famous theorem is on the finiteness of the form-system of one or more binary forms. After a slight introduction, the subject is discussed under six heads, viz. Mertens' demonstration and Hilbert's first proof of the theorem; Hilbert's general proof for forms in n variables Deruyts' researches in enumeration of covariants of given characteristics; Hilbert's theorem upon syzygies of higher orders; miscellaneous topics; and the writer winds up with desiderata and remarks upon courses of instruction. It will be gathered from the above selection of headings that the report is likely to be useful to students. There are numerous references, which we hope are more accurate than the following footnote on the first page, which cites Sylvester's proof of the theorem; thus “Proc. Lond. Math. Soc, vol. 27 (1878), p. 11-13.” There is a remarkable muddle here. Vol. xxvii. gives Proceedings of Session 1895-6, the Proceedings of 1878 appear partly in vol. ix. and partly in vol. x., and no reference to such communication occurs in the index to the first twenty volumes of the Proceedings.—Miss C. A. Scott discusses Holgate's translation of Reye's “Geometrie der Lage.” —Prof. M. Bôcher gives an account of Burkhardt's “Funktionentheoretische Vorlesungen,” vol. i., einführung in die Theorie der analytischen Functionen einer complexen Veränderlichen. The little work is said to be a useful introduction not merely to those parts of the theory which have been long classical, but also to the many other important developments of the last thirty years.—An extensive review of Darboux's “Leçons sur les Systèmes orthogonaux et les coordonnées Curvilignes,” by Prof. E. O. Lovett, which follows, bristles with references to original memoirs.—Prof. J. Pierpont warmly commends the new Mathematical Encyclopædia, the success of which he regards as being mainly due to the genius, energy and courage of a single man, Felix Klein.—Errata, Notes (which are fuller than ever under Prof. Lovett's care), and new publications close the number.