126 REPORT — 1900. 



By a generalisation of simple cases, this number of solutions is 

 reduced to the following formula : — 



(A) {k+i),=[{k+i-iuk),,]-[{k+i-2%{k-i),;\+ . . . 



. . . ::^[{k),(k-i + l\]±{k-i% 



where [{k + i — l)k{k)^] stands for the number of common solutions of 

 ||yi; + i — l|j,.. (equivalent to i independent equations, provided the k — 1- 

 rowed determinants, mentioned above, do not vanish), and of \\k\\,, which 

 represents precisely one equation ; and so on. A rigorous proof of this 

 formula was not given until 1890,^ but, assuming it to hold, the number 

 of solutions, when the variables are all independent, is found by perfectly 

 valid reasoning in the paper under consideration, and particular cases of 

 the more general problem, to which formula (A) also applies [i.e. when 

 there are k pairs of variables, connected by k relations), are solved in 

 the next paper - by direct evaluation. 



When the terms of the right-hand side of (A) come to be actually 

 evaluated, the particular case, here alone considered (i.e. of k independent 

 variables), proves capable of direct treatment by algebraic theorems in 

 elimination proved in the earlier part of the paper, and the final result is 



/7 . -v (w^ — ^ + l)(m — k) . . . (m—k — i + 1) 



^^+'^^ ITTVsTTTi+l 



From the point of view of the theory of point-groups, a geometri- 

 cal problem which Brill solves by means of formula (A) is of interest ; it 

 is thus stated : 



Given a {k + i — ^)-ply infinite family of curves of order m, viz. : 

 a,0](x,y) + a2^2(x,y)+ ;•• +f';.+i0i+i(x,y)=O. Assuming k-i-1 of the 

 points of intersection with a straight line, to find i others such that every 

 curve through these k— 1 also passes through a cei-tain kth point. In how 

 many ivays can this be done ? 



Or, in other words : 



In hov) many loays can k points he chosen on a straight line, so that an 

 i-ply iifinite system of curves, selected from a (k + i — ^)-p>ly infinite system 

 may pass through them ? 



Since the number of solutions is all that is required, the problem is not 

 made less general by taking the intersections with a definite straight line, 

 say y=0 ; substituting this value for y in the equation from the outset, 

 we are led to finding the number of solutions of exactly the matrix con- 

 sidered on p. 125, leading to formula (A), which, since a;,, . . . x,. are k 

 independent variables, can be directly evaluated as above. . 



Brill's investigations into the theory of correspondences definitely 

 commenced in 1872.'' In the introductory remarks he attributes the 

 origin of this theory (in geometry) to Chasles, who, in 1864, first enun- 

 ciated the principle of correspondence for points on a straight line : 

 ' if to every point x there are n points y, and to every p)oint y tliere are m 

 points X, tlben at m + n points an x coincides with ay;' and who af ter- 

 wai-ds extended it, in 1866, to points on any unicursal curve.^ Cay ley 



' Math. Ami., vol. xxxvi., p. 326. = Ibid., vol. vi. 



3 Ihid., vol. vi., pp. .33-65, ' Ueber Entsprechen von Punktsj'stemen auf einer 



Curve.' 



* Comptes Rendus, vol. Iviii., June 27, 1864, and vol. Ixil., p. 11. 



