and the general theory of Algebraic Curves. 81 



studied upon the cardinal curve of the group. Equivalent sets 

 are cut out from the cardinal curve by surfaces of the same order, 

 having certain common intersections with that curve. 



Prop. XII. To any group of R points of multiplicity r, 

 corresponds on the cardinal curve a group of R points which lie 

 in a space of R — r-1 dimensions. For ordinary sets of points 

 this tells us nothing, inasmuch as R — r = p, and the cardinal 

 curve lies in space of p — 1 dimensions : for special sets of points 

 it leads at once to the Riemann-Roch theorem. 



These propositions follow almost at once from Humbert's re- 

 searches : some of them are known, and will be found in Segre's 

 writings (Math. Ann. xxx. p. 203, and elsewhere). But the 

 simplicity of the proofs by automorphic functions is remarkable. 



VOL. XII. PT. II. 



