( 108 ) 



(Mii've) and tlio (iiioslioii aiiiiox to it after the singularities ot' the 

 evolnte. It seems to nic tluil in no ollici- \va_v knoNNn to me tiiese 

 and sneldike (juestions can Ije so simpiv answered. 



The j^enus lias heeii introduced in the theory of functions by 

 RiEMANN and is delin(>d out of the coiniectiou of the ?/-lea\'ed RiKMANN- 

 surface on which an //'-xalucd al,L»('l>raic funclion is uuixalejit. If .v 

 is the numher of l)i-anch|»oints of the function, ƒ/ the genus, then 

 we have the following i-elation given i)y Rikmann ^) (I.e. p. 129) 



•S— 2 n = •2{;l~-i), 



for which a i)ranchi)oint where / leaves of the lviiv\L\NN-surface are 

 connected is to be regarded as t — 1 bi-anchpoints. 



Foi- llie theory of the algebraic cur\ cs the notion and also the name of 

 genus ("Gescidecht") has been introduced by Clküsch'), whilst Hai.phkn') 

 lias given for the genus of a curve of order /a and class k with 

 higher singularities the equation 



in which JS" {ti — 1) represents a summation over the origins of the 

 separate branches of the curve (which can I)e re])resented by one 

 PuisKLX-development) whose order t ditfers from 1 and over as many 

 other origins of bi-anches as one likes. If a branch of the curve is 

 represented by the development 



y — n = ^0 (•'' — §) + ^'i (•'• — ») + • • • 1 



I 



according to integral ascending powers of (./; — i,) , 1 call the point 

 (5 , rj) the oi'ujin., the line // — i^ =: ^/„ (./• — 5) the tnncfent and the 

 luimbei'S t and v the ord'r and the rA/.s.v of the bi-anch, where thus any 

 point of a curve can be regarded as the origin of at least one 

 branch, for which however if the point is a common jxjint of the 

 curve / will be equal to 1. If one and the' same ])oint of the curve 

 is the origin of more branches we shall regard this point successively 

 as if belonging to the different branches. 



The HALPHEX-relation is an immediate result of the Riemann- 

 relation if only one decomposes the branchpoints into those which 



1) B. RiEMANN. Theorie der ABEL'schen Functionen. Crelle's Journal, vol. 04, 

 (1857), p. 115—155. 



') A. Glebsch. Ueber die Singulariltiten algebraisclier Curven. Crelle's Journal, 

 vol 64, (1865), p. 98-100. 



^) G. H. Halphen. Sur la conservation du genre des courbes algébriques dans 

 les transformations uniformes. Bulletin de la Soc. Moth, de France, vol. 4, (ISTö), 

 p. 29-41. 



