Scientific Serials

Abstract

Transactions of the American Mathematical Society vol. iii No. 1, January.—On a class of automorphic functions, by J. L. Hutchinson. In Burkhardt's “Ueber die darstellung einiger falle der automorphen primformen durch Specielle Thetareihen,” the following monodromy group of the Riemann surface, y3 = (x-a1)(x - a2) (x-β1)2 (x - β2)2, is considered, and he shows how a certain prime form which is automorphic for the group can be expressed by a theta series. Further results are here given concerning the group and the functions belonging to it, the chief object being to obtain explicit analytic formulæ by means of which all functions of the group can be represented. To this end the theta-fuchsian functions of Poincare are introduced, and their expressions in terms of the hyperelliptic theta series deduced.—Concerning the existence of surfaces capable of conformal representation upon the plane in such a manner that geodetic lines are represented by a prescribed system of curves, by H. F. Stecker, is in continuation of a previous paper under nearly the same title (vol. ii. p. 152).—Zur erklärung der Bogenlaönge und der inhaltes einer Krummen flaöche, by O. Stoll (cf. the author's “Gründziige der Differential—und Integralrechnung,” Bd. 2, and Math. Ann., Bd: 18).—The groups of Steiner in problems of contact, by Dr. L. E. Dickson, gives an elementary proof Nof Jordan's (“Traité,” pp. 229–249). Reference is given to Steiner and Hesse (Journal für Math., vol. xlix. (1855) and vol: lxiii. (1864), and to papers by the author {Bulletin Amer. Math. Soc., vol. iv., and the American Journal of Mathematics, vol. xxiii. pp. 337-377).—Quaternion space, by A. S. Hathaway, follows up Stringham's work in vol. ii. p. 183, but frequent reference is made to Clifford's paper on biquaternions. Stringham deals analytically with the equations of loci and develops the geometry by the interpretation of those equations; the author uses a more synthetic method, which interprets the quaternion symbols themselves instead of the equations between them. It is this divergence which constitutes the general difference between the methods of Cayley and Tait. Clifford stated the synthetic view in his further note on biquaternions.—Reciprocal systems of linear differential equations, by E. J. Wilczynski, arrives at interesting results in connection with previous papers (Transactions, vol. ii. No. 4; American Journal of Mathematics, vol. xxiii.).—On the invariants of quadratic differential forms, by C. N. Haskins, investigates, by means of Lie's theory of continuous groups, the problem of determining the number of invariants of the general quadratic form in n variables. Numerous references occur in the paper.—On the nature and use of the functions employed in the recognition of quadratic residues, by, Dr. E. McClintock, refers to Tannery, “Lecons d'Arithmétique,” Bachmann, “Elemente der Zahlentheorie,” and to Baumgart,“Ueber der Quadratische Reciprocitätsgezetz.”—A determination of the number of real and imaginary roots of the hypergeometric series, by E. B. Van Vleck. Concisely we must refer to Klein (Math. Ann., vol. xxxvii. p. 573) for the number of the roots of the equation considered between o and —α. Mr. Van Vleck claims that he gives, for the first time, the number of imaginary roots. Numerous references and diagrams (six and a page of sixteen) accompany the text.—The second variation of, a definite integral when one end-point is variable, by G. A. Bliss. The method which the author applies to the discussion of the case in which one end-point moves on a fixed curve is closely analogous to that of Weierstrass (“Lectures on the Calculus of Variations,” 1879). In the present case terms outside of the integral sign are taken into consideration. Then, as a result of the discussion, the analogue of Jacobi's criterion is derived, defining, apparently in a new way, the critical point (Kneser's “Brennpunkt”) for the fixed curve along which the end-point varies. Then the relation between the critical and conjugate points is discussed.—On the projective axioms of geometry, by E. H. Moore, contains a consideration of the axioms called by Hilbert (“Grundlagen der Geotnetrie”) the axioms of connection and of order, and by Schur (“Über die Grundlagen der Geometrie”) the projective axioms of geometry. There are several citations of authorities, such as Peano, Pasch and Ingrami.