TRANSACTIONS OF THE American Mathematical Society EDITED BY EEIAKIM HASTINGS MOORE EBNEST WILLIAM BROWN THOMAS SCOTT FISKE WITH THE COOPERATION OF EDWARD BURR VAN VLECK HENRY SEELY WHITE CHARLES LEONARD BOUTON LEONARD EUGENE DICKSON JOHN IRWIN HUTCH1NSON EDWARD KASNER EDWIN BIDWELL WILSON Published quarterly hy the Society with the support of Harvard University Northwestern University Yale University Cornell University Columbia University The Universität of California Wesleyan University Bryn Mawr College Haverford College The University' of Chicago Volume 5 1904 Lancaster, Pa., and New York THE MACMILEAN COMPANY Agents for the Society 1901 Reprinted with the permission of The American Mathematical Society Johnson Reprint Corporation 111 Fifth Avenue, New York 3, N. Y. Johnson Reprint Company Limited Berkeley Square House, London, W. 1 First Reprinting, 1963, Johnson Reprint Corporation TAELE OF CONTENTS VOLUME 5 1904. PAGES Blichfeldt, H. F., of Stanford University, Cal. On the order of linear homogeneous groups (second paper).310-325 _ A theorem concerning the invariants of linear homo- geneous groups, with some applications to substitution groups 461-466 Bliss, G. A., of Columbia, Mo. An existence theorem for a differ- ential equation of the second order, with an application to the calculus of variations.113-125 _Sufficient condition for a minimum with respect to one-sided variations.477-492 Brown, E. W., of Haverford, Pa. On the smaller perturbations of the lunar arguments.279-287 Cole, F. N., of New York, N. Y. The groups of order fq» . 214-219 Dickson, L. E., of Chicago, 111. The subgroups of order a power of 2 of the simple quinary orthogonal groups in the Galois field of ordery = 8Z±3. 1-38 _ Determination of all the subgroups of the known sim- ple group of order 25920.126-166 Eisenhart, L. P., of Princeton, N. J. Three particular systems of lines on a surface.421*437 Epsteen, S., of Chicago, 111. On the definition of reducible hyper- complex number systems.105-109 Findlay, W., of New York, N. Y. The Sylow subgroups of the symmetric group.263-278 Fréchet, M., of Paris, France. Sur les opérations linéaires . 493-499 Goursat, E., of Paris, France. A simple proof of a theorem in the calculus of variations. (Extract from a letter to Mr. W. F. Osgood.).110-112 Haskins, C. N., of Ithaca, N. Y. On the invariants of quadratic differential forms, II.167-192 Hun, J. G., of Princeton, N. J. On certain invariants of two tri- angleso . 39- 55 Huntington, E. V., of Cambridge, Mass. Sets of independent postulates for the algebra of logic.288-309 Hutchinson, J. I., of Ithaca, N. Y. On the automorphic functions of the group (0, 3; 2, 6, 6).447-460 iv TABLE OF CONTENTS PAGES Kasner, E., of New York, N. Y. Isothermal systems of geodesies. 56- 60 Loewy, A., of Freiburg, i. B., Germany. Zur Gruppentheorie mit Anwendungen auf die Theorie der linearen homogenen Differen- tialgleichungen .,. 61-80 Macaulay, F. S., of London, England. On a method of dealing with the intersections of plane curves.385—410 Mason, Max, of New Haven, Conn. Green's theorem and Green's functions for certain systems of differential equations . 220—225 Morley, F., of Baltimore, Md. On the geometry whose element is the 3-point of a plane.467-476 Peirce, J. M., of Cambridge, Mass. On certain complete systems of quaternion dxpressions, and on the removal of metric limita- tions from the calculus of quaternions.411-420 Rietz, H. L., of Urbana, 111. On groups in which certain commu- tative operations are conjugate.500—508 Boe, E. D., Jr., of Syracuse, N. Y. On the coefficients in the prod- uct of an alternant and a symmetric function.193-213 Shaw, J. B., of Decatur, 111. Algebras denned by finite groups . 326-342 Taber, H., of Worcester, Mass. On hypercomplex number systems (first paper).509-548 Van Vleck, E. B., of Middletown, Conn. The convergence of alge- braic continued fractions whose coefficients have limiting values. 253-262 Veblen, O., of Chicago, 111. A system of axioms for geometry . 343-384 Wilczynski, E. J., of Berkeley, Cal. Studies in the general theory of ruled surfaces.226-252 _ On ruled surfaces whose flecnode curve intersects every generator in two coincident points.438-446 Young, J. W., of Evanston, 111. On the group of the sign (0,3; 2,4, oo) and the functions belonging to it.81-104 Notes and errata : volumes 1, 3, 4, 5.549-552 INDEX BY AUTHOKS. Vol.* Pages. Class.! Allardice, K. E., On the envelope of the axes of a sys- tem of conies passing through three fixed points. 4, 103-106 F 2 Blake, E. M., Two plane movements generating quartic scrolls. 14 421-429 F5 Blichfeldt, H. F., Note on the functions of the form f(x) = <?(x) + axx"~l + a2x"~2 + ... + aMwhich in a given interval differ the least possible from zero. 2, 100-102 D1 -A new determination of the primitive continuous groups in two variables. 23 249-258 E3 -On the determination of the distance between two 467-481 points in space of n dimensions. ÍAI FI -On the order of linear homogeneous groups. 44 387-397 E2 -On the order of linear homogeneous groups (second paper.). 5, 310-325 E2 ■-A theorem concerning the invariants of linear homo- geneous groups, with some applications to substitu- tion groups. 54 461-466 E2 Bliss, Gilbert Ames, The second variation of a definite integral when one end-point is variable. 3, 132-141 D7 -An existence theorem for a differential equation of the second order, with an application to the calculus I D4 *" of variations. 2 1 D 7 -Sufficient condition for a minimum with respect to one-sided variations. 54 447-492 D7 Bôchek, Maxime, On regular singular points of linear differential equations of the second order whose coëffi- f 1, 40- 52 D 4 cients are not necessarily analytic.1 14 507 -Application of a method of D'Alembert to the proof of Sturm's theorems of comparison. 14 414-420 D4 -Certain cases in which the vanishing of the Wron- skian is a sufficient condition for linear dependence.. 22 139-149 Dl -An elementary proof of a theorem of Sturm. 22 150-151 D 4 * The notations of this column indicate for every paper the volume and part of volume of publication, and thus the date of publication, e. </., the first paper in the list was published in January, 1903, and the second in October, 1900. In this column references are made also to the Notes and errata published in the concluding numbers of various volumes, e. g., a note to the twelfth paper appears in the Notes and errata of volume 1. fThe notations in this column refer to the Index by Subject Matter. 5 6 INDEX BY AUTHORS [October -On certain pairs of transcendental functions whose roots separate each other. 2. 428-436 D4 -On the real solutions of systems of two homogeneous linear differential equations of the first order. 196-215 D4 Bolza, Oskar, The elliptic «--functions considered as a 53- 65 D3 special case of the hyperelliptic «--functions_. 484 -New proof of a theorem of Osgood's in the calculus of variations. 2, 422-427 D 7 -Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so- called isoperimetric problems. 33 305-311 D7 Bromwich, T. J. I'a., Similar conies through three points 489-492 F 2 Brown, Ernest AVilliam, On the small divisors in the 1unar theory. 159-185 G 2 -On the formation of the derivatives of the lunar co- ordinates with respect to the elements. 4. 234-248 G 2 -On the variation of the arbitrary and constants Gl given 333-350 in dynamical equations. G2 279-287 G 2 -On the smaller of the lunar arguments perturbations 551 Cajori, Florian, Divergent and conditionally convergent series whose product is absolutely convergent. 25- 36 Dl Campbell, J. E., On the types of linear partial differen- tial equations of the second order in three inde- variables which are unaltered by the trans- 243-258 D 5 pendent j formations of a continuous group. 509 1 E 3 Chessin, Alexanders., On relative motion. 116-169 G1 Coble, A. B., The quartic curve as related to conies. 65- 85 F 2 Cole, F. N., The groups of order ps<f. 214-219 E1 Coolidge, Julian Lowell, A purely geometric represen- tation of all points in the projective plane. 182-192 FI -■ Quadric surfaces in hyperbolic space. 161-170 FI Darwin, G. H., The approximate determination of the form of Maclaurin's spheroid. 4, 113-133 G 3 Dickson, Leonard Eugene, Definition of the abelian, the two hypoabelian, and related linear groups, as quo- tient groups of the groups of isomorphisms of certain elementary groups. 30- 38 E2 -A new definition of the general abelian linear group.. 91- 96 E2 -Determination of an abstract simple group of order 2'-36 5-7 holoedrically isomorphic with a certain orthogonal group and with a certain hyperabelian 353-370 E 2 group. 509 -Canonical forms of quaternary abelian substitutions 103-138 E 2 in an arbitrary Galois field. 499 1904] INDEX BY AUTHORS 7 • 363-394 E2 -Theorymi of«it linear in an arbitraryu*~ *field.i m í2< groups (. o. 000 C Ql S8— 45 F 2 -The groups of Steiner in problems of contact.\ -On the group defined for any given field by the multi- plication table of any given finite group. 33 285-301 E 2 -The groups of Steiner in problems of contact (second paper).