ON THE PRESENT STATE OF THE THEORY OF POINT-GROUPS. 123 



(v.) Brill and Noether's report on the theory of algebraic functions, 

 containing succinct accounts of the contents and iraportance of many of 

 the memoirs in the above divisions. 



(vi.) Solution of the question of the identity of the terms involution 

 and linear series by Humbert and by Castelnuovo (1893). 



(vii.) F. S. Macaulay's papers in the Proceedings of the London Mathema- 

 tical Society, vols. 26, 29, 31, 1895-99, on curves through given points. 



§ 3. Analysis according to Content. 



A. The three different methods of investigation, viz. analytical, 

 geometrical, transcendental. 



B. Various definitions of the terms in use by English, French, German, 

 and Italian writers, and the logical connection of the ideas when defined 

 in the language of analytical geometry. 



C. Results obtained by the theory, expressed in the terms defined 

 inB. 



(o) Concerning the linear series of point-groups on a given base-curve, 

 e.g., Clifford's theorem, the liiemann-Roch theorem. 



(y8) Concerning the base-curve, proved by means of the properties of 

 linear series of point-groups, and of linear systems of plane curves, e.g. : 



(i.) Persistence under hi-rational transformation of p {deficiency, 

 genus). 



(ii.) Reduction of the order of a curve toith given deficiency. 



(iii.) Classification of plane curves into rational, hyperellij^tic, \i-gonal. 



D. Properties of surfaces in hyperspace in connection with the pro- 

 perties of linear systems. 



§ 4. Beill's Memoirs on Elimination and Algebraic Correspondences. 



1863-1873. 



Brill's earliest papers in the Mathematische Annalen are on problems 

 which arose naturally out of the subject-matter of his Habilitations- 

 schrift, viz. the transformation theory of algebraic functions in connec- 

 tion with Riemann's memoirs on Abelian functions. Clebsch and 

 Gordan, in their treatise on Abelian functions, published in 1866 (the 

 year before Brill's Habilitationsschrift), had attempted to develop 

 a theory of the applications of Abelian functions to geometry. Inter- 

 preting Abel's and Riemann's equations as curves, they expressed 

 the number p, which plays a fundamental part in Riemann's theory of 

 Abelian integrals, in terms of the singularities of the corresponding plane 

 curve, thus identifying it with the number studied by Cayley under the 

 name of deficiency. They further discussed the persistence of this number 

 under bi-rational transformation — that is, the simplest type of a one-one 

 correspondence — and the existence of certain constants (moduli) invariant 

 under such transformation. Brill adopted their interpretation of the 

 equations, and his work, though essentially analytical in form, is capable 

 of direct geometrical application in its results. 



The number of the moduli is the subject of the first two papers. In 

 the earlier of the two ^ he remarks that Riemann, by analysis, found 



' Math. Ann., vol. i., pp. 401-406, 1869, ' Note beziiglich der Zahl der Moduln einer 

 Classe von algebraischen GleichuDgen.' 



