( 107 ) 



certainly excluded as soon as we [)ul a^„f' = Up a,^, f'oi- there is then 

 an {n — 1) fold root ).z=z() and a rool A = oo. 



Resumini»' : 



By assnniijio- tlie rnle of 0.\litzink-1>krthki.ot, we find: 



n=^'2. No niaxirniini ; a niiniiniun is possible. 



71 = 3. No maxinHiin or niininuini — a stationai-y point, hut no 



niaximnni oi- mininnun, is jiossible. 

 n, = 4. No maxininni or niininmin ; other stationary points are 



possible. 

 ;/ = 5 and higher. All stationary points are excluded. 



If we assnme for b a linear function of,/', thus l>ijq^^^l.i{l>,,-\- l>,i), 

 then already for }i, := 3 the determinant on the //s becomes identical 

 to 0, so that then for ternary and higher mixtures no stationary 

 points are any more possible as soon as we put (lp^|^ eipial ti)(/j,<i(j. 



Mathematics. — "On an ('.iipress-ion for the genus of an algebraic 

 plajie curve unlh hig/ier s/.iigu/a)-/t/'es." By Mr. F'red. Schuh. 

 (Communicated by Prof. D. J. Korteweg.) 



Lately I gave the following theorem in these Proceedings ^) : 

 Let P be a point of order t' of an. algebraic plane curve {v)here 

 t' Cdu. also be zero, namely 'when P does not lie on the curve) and 

 S an arbitrarg point of order t of that curve. Suppose the straight line 

 PS intersects the curve in id points coinciding n-if/i S, the// /'-|-^ {u\ — tj) 

 {suiniiied up oeer all points S for u^hich /n is ^ t) is independent 

 of the situMion. of point P and equal to the class of the curve. 

 If S lies in P uje luive to regard all straight lines through P as the 

 line connecting P and S or if one likes only those u^hich furnish a 

 conti'ibution to 2 {n\ — t^) i. e. the tangents in P. 



From this a corresponding and as far as [ can see a more 

 imj)ortant theorem for the genus of an algebi'aic curve can be 

 deduced, where moreover the straight liiu' connecting Pand S caw ha 

 replaced by an algebraic curve. Lateron 1 iioj)e to comiect this with 

 problems of contact fmnnbers of algebraic curves determined by 

 conditions of contact) in particular with respect to the nnml)er of 

 normals on a curve with higher singularities (also i]i connection 

 Avith the circle points and the line at infinity) let (h)\vn from a j)oint 

 (which can also have a j)articular situation with respect to the 



1) On an expression lor tlie class of an algebraic plane enrve with higher 

 singularities. These Proe. Vll, p. 42. 



