The Mathematical Gazette. Edited by F. S. Macauley, St. Paul's School, West Kensington, London. Issued three times a year, viz in February, June, and October. Price, one shilling, net. The June number contains an article on Spherical Geometry: I. Orthogonal Projection, by Prof. Alfred Lodge, M. A.; II. Stereographic Projection, by P. J. Heawood, M. A. Also Notes, Mathematical Notes, Examination Questions and Problems, Solutions, and Reviews and Notices. In "Notes" is an extended notice of Dr. Halsted's article on the "Non-Euclidean Geometry" which appeared in the March number of the MONTHLY. B. F. F. The Monist. A Quarterly Magazine devoted to the Philosophy of Science. Edited by Dr. Paul Carus; T. J. McCormack, Assistant Editor; E. E. Hegeler, and Mary Carus, Associate Editors. Price, $2.00 per year in advance. Single number, 50 cents. The Open Court Publishing Co., Chicago, Ill. The following articles appeared in the January, 1897, number: The Logic of Relatives, by Chas. S. Peirce; Man as a Member of Society, Introduction, by Dr. P. Topinard; The Philosophy of Budhism, by Dr. Paul Carus; Panlogism, by E. Douglas Fawcett; The International Scientific Catalogue, and the Decimal System of Classification, by Thomas J. McCormack; and Literary Correspondence-France, by Lucien Arréat. B. F. F. The American Monthly Review of Reviews. An International Illustrated Monthly Magazine. Edited by Dr. Albert Shaw. Price, $2.50 per year in advance. Single Number, 25 cents. The American Monthly Review of Reviews Co., 13 Astor Place, New York City. We are pleased to note that since our last issue this valuable magazine has changed its name to The American Monthly Review of Reviews, a more significant title than its for mer one. The September number has a good deal to say about the Andrews incident and Brown University-not so much, as the editor remarks, on account of the personal interests involved in the case, as because of the far-reaching principles affecting academic life and liberty which have become matters at issue. A fair-minded and judicious estimate of President Andrews' services to Brown is given by a writer fully conversant with the facts, and the protest of the faculty is printed in full. The editorial comments on the awkwardness and needlessness of the situation are piquant and to the point. Among the contributed articles in the September number are sketches of the three members of the new Nicaragua Canal Commission-Admiral Walker, Capt. O. M. Carter, Corps of Engineers, U. S. A., and Prof. Lewis M. Haupt. These sketches are illustrated with portraits, and serve to convey an idea of the peculiar qualifications possessed by these gentlemen for the task to which they have been appointed by President McKinley. The Arena. An Illustrated Monthly Magazine. Ridpath, LL. D. Price, $2.50 per year in advance. Boston: The Arena Co. B. F. F. Edited by John Clark Single number, 25 cents. Every true American citizen should read Dr. John Clark Ridpath's splendid paper, "The Cry of the Poor," and his "Open Letter" to President E. B. Andrews, which appear in the September number of the Arena. In them the Doctor has drawn a picture that appeals to every man and woman in our land who has God-given rights and privileges which, owing to the intervention of plutocratic influences, they are not allowed to enjoy. "Why," asks the Doctor, "should the voice of the poor ever be heard rising like a wail from plantation, hamlet, and cityful? Why should there be seen standing at the door of the homes of the American people the gaunt spectre-Want ?" "And why," he again asks, "should we allow the voice of our teachers to be smothered by plutocratic powers?" There may be those who sanction the conduct of Brown University in expelling Professor Andrews, but it is very evident that the editor of the Arena and the author of The Bond and the Dollar" and "The True Inwardness of Wall Street" does not. Among the other papers are "The Concentration of Wealth, its Cause and Results: Part I," by Herman E. Taubeneck; "The Multiple Standard for Money," by Eltweed Pomeroy; "The Future of the Democratic Party: A Reply," by David Overmyer; "The Author of "The Messiah'," by B. O. Fowler; "Anticipating the Unearned Increment," by I. W. Hart; "Studies in Ultimate Society:" I. "A New Interpretation of Life," by Laurence Gronlund ; II. “Individualism vs Altruism," by K. T. Takahashi; "General Weyler's Campaign," by Crittenden Marriott; the "Plaza of the Poets," "Book Beviews,” and “The Editor's Evening," make up this bright and instructive number. CORRECTIONS AND REVISIONS OF THE ARTICLE "ON THE CIRCULAR POINTS AT INFINITY,” (P.=page; 1. x=xth line from above; lb. x=xth line from below.) 2 P. 132, 1. 1 of the article, read Coördinate for Coordinate; 1. 2, Cartesian for Cartesion. P. 133, (4) and (4)' for (A) and (4)'; 1. 19-21, finish parenthesis; 1. 23, = for P. 134, 1. 2, vanishes for vanisnes; interchange lines 14 and 15. P. 135, 1. 4, bring "all true" down to 1. 6; 1. 12, add "and" after "infinity"; 1. 19 and 23, coördinates for coöordinates; 1. 25, coördinates for coödinates. P. 136, 1. 4, add exponent 2 to numerator; 1. 6, p2 here taken equal to 1, might have been retained in the numerator. If retained, (21) p. 140 would contain p instead of p2, but this would have no effect on the final result (22). Whether p2 is retained or not, (14) would have to be made homogeneous in all the coördinates involved, as well as (21), for practical uses, since this is required of all such equations. (14) can be made homogeneous by the use of the solution of (4). 1. 9, +sina, sina, for sina, sina,; -x12cosС for x, x cosС; lb. 6, c for C. P. 137, l. 16, r for y. P. 138, lb. 4, X2' '2 for x,'; lb. 5, r for y; lb. 8, cos C for cosB; lb. 9, x,' for x, 2. P. 139, 1. 5, x, for x, and for x,; 1. 12, xx, for x,x,. P. 140, lb. 2, (x'') for (x'). P. 141, 1. 14, x' for x ; l. 17, xqu2+x, uz for qu2+uz; lb. 1, x,'2 for x'; in foot note, "NichtU z 2 Euklidische Geometrie" for "Nicht-Euclidsche Geometry." P. 142, 1. 9, -iA for iB; 1. 11, two lines for the lines; 1. 13 and 14, x and y might be interchanged, though this is not necessary; the other angle between the two lines would be given; l. 15, The double ratio of these is: Taking them in the order named, using etc.; 1. 18, 8 for 5; 1. 19, +sλ' for +sλ. P. 144, l. 11, tany for; slopes for tangents would be better; 1. 12, it is necessary and sufficient that the purely imaginary part of x should become indefinitely great; 1. 18, the German word "quadrupel" is here appropriated; 1. 23, ±1 for ±l; 1. 27, is for in; in foot note, * for †. P. 145, l. 4, Exx. Zx'x' for Zxx'. Ex'x in numerator and denominator; (Zxx')2 for Zxx' under radical in denominator; 1. 5, two points he points; 1. 7, Exx for Exx'. 2 2 2 Աջ 2 2 THE AMERICAN MATHEMATICAL MONTHLY. VOL. IV. Entered at the Post-office at Springfield, Missouri, as Second-class Mail Matter. OCTOBER, 1897. SOPHUS LIE'S TRANFORMATION GROUPS. No. 10. 1. A SERIES OF ELEMENTARY, EXPOSITORY ARTICLES. By EDGAR ODELL LOVETT, Princeton University. I. Without entering unnecessarily into definitions which will occur more properly later, the following paragraph may serve for purposes of orientation. Among the most important notions of modern pure mathematics are the idea of a group and its associated notions transformation, substitution, invariant and differential invariant. Groups fall naturally and historically into two classes, discontinuous and continuous. The former are usually called substitution groups and are not infrequently referred to as GALOIS' groups; the latter are known as continuous transformation groups and may with propriety be called LIE groups. Substitution groups find their greatest usefulness in the theory of algebraic equations, with a limited range of application to geometry; transformation groups play a similar rôle in the theory of differential equations, with a wide application to geometry and mechanics. The idea of a substitution group in its modern signification and in its relation to the theory of algebraic equations is due to GALOIS; LIE, after having modified and extended the idea of a substitution group, introduced the new notion into the domain of analysis and geometry and thus created his theory of transformation groups. The great fruitfulness and remarkable simplicity of LIE's theories are their most striking characteristics. Because of their manifold applications, a thorough and systematic study of the fundamental properties of continuous groups is cer |