Scientific Serial

Abstract

American Journal of Mathematics, vol. xviii. No. 3. (Baltimore, July.)—On the multiplication and involution of semi-convergent series, by Prof. Cajori. In vol. xv. Prof. Cajori has generalised Voss's results (Math. Ann., vol. xxiv. p. 42), and some further contributions of his to this difficult subject are given in the Bulletin of the Am. Math. Soc. (vol. i. pp. 180–183). The search, he remarks, for expeditious tests on the applicability of Cauchy's multiplication rule to powers of semi-convergent series higher than the second power, has given rise to the present investigation, which begins with alternating semi-convergent series, and ends with certain trigonometric series.—Analytic functions suitable to represent substitutions, is an interesting following-up of a theorem due to Hermite (Comptes rendus, vol. lvii. p. 750), by L. E. Dickson. Further generalisations are promised in a dissertation by the author.—S. Kantor contributes an elaborate memoir, “Theorie der Transformationen im Rr, welche sich aus quadratischen zusammensetzen lassen,” which has as heading, “Boldness is caution in these circumstances.”—Tactical Memoranda, i.-iii., by E. H. Moore, is the opening one of a series of papers which the author proposes to publish, on certain more or less closely connected topics of tactic. He starts from Cayley's division of algebra into tactic and logistic. This instalment bears upon the work of Reze (Geometrie der Lage), S. Kantor, Klein, and many others; it also gives a generalisation of the fifteen-schoolgirls arrangement, and considers whist tournament arrangements, which are in ultimate formulation purely tactical.