124 EEPORT— 1900. 



3 p — 3 to be the number of moduli of his ' normal ' function, whereas 

 Cay ley ' obtained, by geometrical considerations, the number 4 ^^ — 6 for the 

 curve of (2; + l)th order, the 'normal' curve of Clebsch and Gordan ; but 

 by actually performing the transformation of the latter — in the case /;=4 

 — into Riemann's form. Brill shows that there are in fact only 3 p — 3 

 moduli (a result which Cayley verified later.) ^ The second paper ^ was 

 occasioned by a memoir by Casorati and Cremona,'' in which the trans- 

 formation of Clebsch and Gordan's form into Riemann's is effected, by 

 geometrical methods, for the cases jo=4, 5, 6. Brill obtains their results 

 by different methods, employing the properties of curves in space of three 

 dimensions. An example for p = 6is3i septic with nine double points ; this 

 he connects by a one-one transformation with a curve in space of the 8th 

 degree, quoting Cayley ^ to show that the transformation can be effected 

 in exactly five ways, corresponding to the five straight lines in space 

 which meet the tortuous curve of the 8th degree in four points. Similarly 

 for^^^T, the transformation of a plane curve of the 8th degree can be 

 effected in twenty-one different ways. These examples are important, as 

 forming a connecting link between the theory of transformation, in which 

 they presented themselves, and the theory of elimination, to which they 

 directly lead ; moreover, within the theory of elimination they suggest 

 the question of the number of different solutions satisfying a system of 

 simultaneous equations. 



In the year 1871, Brill begins to turn his attention to a wider theory, 

 that of elimination when stated algebraically, or of correspondences when 

 stated geometrically. This is shown in the title of a paper,^ which con- 

 tains proof of theorems required in the succeeding paper ; '^ but neither 

 of these has any immediate application to our present purpose. 



The geometrical side of the theory of correspondences had been 

 already attacked by Chasles, De Jonquieres, and Cayley, but algebraical 

 proofs of many theorems were still wanting ; and, moreover, the treat- 

 ment of the problems in a purely symbolical and analytical manner led 

 to the establishment of theorems in the general theory of elimination, 

 which in their turn apply to a region intimately connected with the 

 theory of correspondences — that of point-groups on a curve — but at the 

 date we speak of, still comparatively unexplored. 



In Brill's first important contribution to the theory of elimination,*' 

 he attacks the problem of the number of different solutions which satisfy 

 a system of simultaneous equations.^ He remarks that Roberts '" and 

 Salmon '^ confined themselves to a discussion of the degree of the 

 eliminant in the whole number of variables, not the degree in which 



■ Proc. London Math. Soc, vol. i., 18G5, ' On the transformation of plane curves.' 



- Math. Ann., vol. viii., pp. 359-362. ' On the group of points G^ on a sextic 

 curve with five double points,' 1874. 



^ Ibid., vol. ii., pp. 471-474, 1870, ' Zweite Note beziiglich der Moduln einer 

 Classe von algebraischen Gleichungen.' 



■* Accad. Milan, May. 1870. ^ Phil. Trans., 1870, ' On skew surfaces.' 



^ Math. Ann., vol. iv., pp. 510-526, 'Zur Theorie der Elimination und der 

 algebraischen Curven.' 



' Ibid., pp. 527-549, ' Ueber zwei Beriihrungsprobleme.' 



* Math. Ann., vol. v., pp. 378-396, 1872, ' Ueber Elimination aus einem gewissen 

 System von Gleichungen.' 



^ See also Math. Ann., vol. iv., pp. 542-548, 1871. 



'° Crelle, vol. Ixvii., pp. 266-278, 1867. 



" Higher Algebra, Lessons VllI. and XVIII. 



