and the general theory of Algebraic Curves. 79 



[These theorems express in geometric form the fundamental 

 results of Poincare stated in Forsyth's Functions, § 310 ; the latter 

 is the extension of Cay ley's 'Cone and Monoid' method for space- 

 curves.] 



For two reasons the most important curves of a group are the 

 Cardinal curve and the Essential curves now defined : — 



Def. The Cardinal Curve of a group of genus p is the curve 

 of order 2p — 2 belonging to space of p — 1 dimensions defined 

 by the fact that the coordinates of its points are proportional 

 to p linearly independent holomorphic thetafuchsian functions 

 of the first degree. 



Def. The Essential Curves of a group are curves of order 

 m=Zfjb(p— 1), where //, = 2, 3, 4 ..., each belonging to space of 

 m—p dimensions, and denned by the fact that the coordinates 

 of its points are proportional to linearly independent holomorphic 

 thetafuchsian functions of degree fi. 



Prop. III. The Cardinal curve and the Essential curves of 

 a group, (i) are the only curves of a group on which geometry 

 can illustrate the properties of the group unaffected by irrelevant 

 matters ; and (ii) from them every curve of the group may be 

 derived by projection. 



Def. Special curves of a group are those whose points have 

 coordinates proportional to holomorphic thetafuchsian functions 

 of the first degree ; non-special curves of the group are those for 

 which the coordinates cannot be so expressed. 



[This necessary distinction is made by Humbert for plane 

 curves: as the second kind of curves is defined by a merely 

 negative property the word 'non-special' is preferred to Humbert's 

 word ' normal.'] 



Prop. IV. The cardinal curve is the chief of the special 

 curves of a group. All the other special curves of a group may 

 be derived from it by projection, and every curve of the group 

 derived from the cardinal curve by projection is a special curve. 



Prop. V. The cardinal curve is the unique curve of the group 

 of order 2p - 2 in space of p — 1 dimensions. If p < 3, there is 

 no cardinal curve ; if p = 3 the cardinal curve is a plane quartic ; 

 if p = 4, it is the curve of intersection of a conicoid and a cubic 

 surface in a space of three dimensions ; if p = 5, it lies in space 

 of four dimensions and is the curve common to three surfaces 1 

 of the second order. 



1 By the word 'surface' is understood a locus defined by a single homogeneous 

 equation among the coordinates of its points ; should the equation be of the first 

 degree the locus is called a 'plane,' so that in space of d dimensions a 'plane' is 

 a space of d - 1 dimensions. 



