122 REPORT— 1900. 



means of homogeyieous co-ordinates; ' and also to fix one curve in the 

 discussion, treating the other curves as variable.- Cayley's theorems are 

 interesting on account of the subsequent discussion as to their true 

 formidation.^ 



C. 1857-1873. (i.) Memoirs on bi-rational transformation. 



(ii.) Brill's memoirs on elimination and algebraic correspondences, 

 (1863-1873). [^ detailed account of these is given in § 4, infra.] 



(iii.) Memoirs and other publications connecting the theory of 

 functions with the theory of plane algebraic curves, including those of 

 Clebsch and Gordan, Brill and Noether. 



[Clebsch and Gordan's treatise attempted to found Riemann's residta in 

 the theory of Abelian functions on an algebraic basis: the standpoint is 

 mainly that of projective geometry. To Brill and Noether is due the 

 initiation of the main line of enquiry in the theory of linear series of point- 

 groups on a base-curve, from, the standpoint of bi-rational transformation.^ 



(iv.) Memoirs on the intersections of curves. 



D. 1873-1890. (i.) Noether's memoirs published in the Mathematische 

 Artnalen, and in Crelle, on the theory of functions, and on analytical 

 geometry. 



(ii.) .Memoirs on linear systems of plane curves, treated analytically 

 from the standpoint of bi-rational transformation. 



[These are chiefly by Italian writers, beginning with Caporali in 



(iii.) Castelnuovo's memoirs on linear series of point-groups on plane 

 curves, treated geometrically from the standpoint of bi-rational transforma- 

 tion. 



(iv.) Segre's memoirs on curves and surfaces in hyperspace. 



(v.) Intersection theorems as treated by Bacharach and Zeuthen. 



[These connect Cayley's theorems with Brill and Noether' s theorem of 

 residtcation.] 



(vi.) Brill's memoirs on algebraic correspondences in the Mathema- 

 tische Annalen contrasted with Castelnuovo's memoir on the number 

 of rational involutions to be found on a curve of given genus (deficiency). 



E. 1890-1900. (i.) Castelnuovo's memoir on linear systems of plane 

 curves as determined by given points (1891, 3fem. Torino, vol. 42). 



(ii.) Memoirs on the theory of algebraic surfaces chiefly by Castel- 

 nuovo and Enriques, and summarised by them in an article in vol. 48 

 of the Mathematische Annalen (1896). 



(iii.) Segre's paper on the geometry on a simply infinite algebraic 

 manifold, in which the properties and applications of linear series are 

 derived from theorems in the geometry of hyperspace (Annali di Mat., 

 vol. 22, 1894). >. y yf f \ 



(iv.) Bertini's account of the principal theorems concerning linear series 

 of point-groups on a plane curve, written chiefly from Brill and Noether's 

 standpoint (Annali di Mat., vol. 22, 1894). 



' Cf. Brill and Noether, Jahreaher. d. Beutschen Math. Ver., vol. iii., p. 207, 1894. 

 ' Cf. Brill and Noether, ibid., p. 290. 

 ' Cf. Brill and Noether, loo. ait., p. 545. 



