130 



REPORT — 1900. 



involving jo, it does not exist when j»=0, i.e. when the base-curve is a 

 straight line or a uuicursal curve, and it was for this reason that the 

 actual solution of the pi-evious problem (p. 126) was possible. The 

 required formulae for i^^l, 2, 3 (leading to 2, 3, and 4 simultaneous 

 correspondences), are worked out in the earlier part of this paper for all 

 possible different groupings of the sets, and the final results for examples 

 (a) and {b) are 



, . M-2.M-3.M-4 .„ .. 



(«) TOO -^(M-4) 



(b) 



1.2.3 



M-3.M-4.M-5.M-6 

 1.2.3.4 



M-6.M- 

 1.2 



ly+?Oi^) 



where, as before, M denotes the number of movable points of intersec- 

 tion of each curve of the given system withy*=0. We notice in passing 

 that these results agree with that of p. 126, when^j=0, M=m. 



A. problem in the theory of point-groups, of which the above are par- 

 ticular cases, was first enunciated in the most general form in a paper by 

 Brill and Noether, in 1873.' They state it thus :— 



Given a t-ply infinite system of adjoint curves — that is, of curves 

 passing s — 1 times through every s,- fold point ofi=:.0 — it is required to find 

 Ji points on the base-curve, f^O, ivhichform such a point-group — or set of 

 points — that the curves of the given system ivhich pass through it form a 

 (\-ply infinite system. If the equation of the system is 



this problem leads, by known theorems, to finding the common solutions 

 of all the (^ — ^-^-l) rowed determinants of the matrix. 



<i>i{x2y-2)> ■ 



hi^Rys.), 



where 



are connected by the equations 

 f{x,y,)=0, . . 



?'t+ 1(^2^2) 



fi^ii^uVn) 



^RyR> 



/(ajRyR)=o. 



The simplest case is therefore to be found by taking R=<— 9 + I, and 

 this is, in fact, the only case completely solved. The formula for the 

 number of solutions was given in this memoir, viz. : — 



<^> '(-)|(|)(«^'-') . . . ieven 

 !<-)n'(|(5 + l)) 'Odd 



+ ■ 



' Math. Ann. vol. vii., pp. 269-310. ' Ueber die algebraischen Functionen und 

 ihre Anwendung in der Geometrie.' 



