ON THE PRESENT STATE OF THE THEORY OP fOlNT-GROUPS. 121 



Report on the Present State of the Theory of Pohit^groiijis . — -Part I. 

 Bij Frances Hardcastle, Cambridge. 



Contents. 



PAfiM 



§ 1. Introduction 121 



§ 2. Historical Outline 121 



§ 3. Analysis of the Subject according to Content. ...... 123 



§ 4. Brill's Memoirs on Elimination and Algebraic Correspondences, 1863-1873. 123 



§ 1. Introduction. 



The term point-group is a dii^ect translation of the German word Punkt- 

 grujype, first used by Brill and Noether in the year 1873 in their classic 

 memoir on algebraic functions,' but to my knowledge, although more than 

 a quarter of a century has elapsed since then, there has been no very 

 systematic attempt to present the theory of point-groups to English 

 readers along any of its lines of development. And yet it should prove 

 of interest even to those mathematicians who do not desire to specialise 

 in it, for, historically and logically, it touches upon many distinct branches 

 of pure mathematics. To mention only those which are most directly 

 brought into connection with each other, we have the intersections of 

 plane curves, the elimination of variables from systems of equations, the 

 algebraic theory of correspondences on a plane curve, properties of linear 

 systems of plane curves, and applications of the theory of functions to 

 the theory of curves and surfaces in space of any number of dimen- 

 sions. 



As frequently happens when the progress of a subject has been due 

 to many different writers, the logical and the chronological divisions do 

 not coincide. I have therefore in view a dual arrangement of the 

 subject-matter. In the present instalment of my Report, I have 

 attempted to sketch this proposed arrangement under its two aspects, 

 viz- : as an historical outline (§ 2), and as an analysis according to 

 content (§ 3). This is followed (§ 4) by a detailed account of one of the 

 historical divisions. I hope in the subsequent portions of the Report to 

 deal in a somewhat similar way with the remaining divisions, and to 

 append a complete bibliography. 



§ 2. Historical Outline. 



A. 1720-1818. Memoirs on the intersections of plane curves from 

 Maclaurin to Lame. 



\^Lame was the first to express the linearity of the system of curves through 

 the intersections of two given curves.-^ 



B. 1818-1857. Memoirs and other published accounts of theorems on 

 the intersections of curves from Lame to Riemann, including those of 

 Pliicker and Cayley. 



[Pliicker was the first to introduce explicitly 2^rojective methods by 



' ' Ueber die algebraischen Functionen und ihre Anwendung in der Geonaetrie,' 

 Math. Ann., vol. vii., pp. 269-310. 



* Cf. C. A. Scott, Bull. Am. Math. Soc, vol. iv., p. 262, 1898. 



