( 428 ) 



times that Cs and Cr have besides yls< another movable point of inter- 

 section, being at the same time movable point of intersection of G and 

 Cr- Here is included the case in which this second point of inter- 

 section coincides with Agt , thus where the cui-ves (1- and Ci touch 

 Cr in Ast ; then only one movable point of intersection of Cs and 

 Ct still coincides with Ast, vv^hilst there need be no other movable 

 point of intersection lying on Cr, so that in this way we get no 

 pair of points furnishing a branch of L passing through Agt . So 

 the point Agt is an (i^s -{- rt — /? — y — 3)-fold point of L. 



To determine the multiplicity of a point Arst we have to consider 

 how many times three curves Cr, Cs and Ct touching each other 

 in Arst pass once more through a same point. To this end we con- 

 sider an arbitrary Cr and the Cs which touches this Cr in Arst- 

 Through each of the rs — y — 1 points of intersection of these C,- and 

 Cs, differing from the basepoints, we allow a Ct to pass. Then the 

 question arises how many times this 6^^ touches Cr and Cs in Arst- 

 Let us call Irs the common tangent in Arst of Cr and Cs and It the 

 tangent of Ct in that point. To 4s correspond 7's — y — 1 lines 

 It. To find reversely how many lines Irs correspond to an arbi- 

 trary line It we consider an arbitrary Cr intei'secting the C deter- 

 mined by It in rt — ^ points differing from the basepoints. Through 

 each of those points of intersection we imagine a Cg. If 4 and I 

 are the tangents in .4,.,^ of Cr and Cs then rt — i? lines 4 corre- 

 spond to 4 and ó'^ — « lines 4 to 4. The rt -\- st — a — /? rays of 

 coincidence indicate the lines 4s corresponding to It; to those rays 

 of coincidence however belongs the line It itself, which must 

 not be counted, so that rt-\-st — a — j3 — 1 lines 4s corresponding to 

 It remain. So between the lines 4s and It exists an {rs — y — i,rt-\- 

 -|- st — « — ^ — l)-correspondence. 



The required lines 4s< are indicated by the st-\-tr-\-rs — («-|-/?-|-y) — 2 

 rays of coincidence of this correspondence of which however three 

 must not be counted. When namely the contact in Arst of Cr and 

 Cs becomes a contact of the second order one of tlie?'5 — y — 1 points 

 of intersection differing in general from the basepoints of Cr and Cs 

 coincides with Arst, namely in the direction of 4.< • The Ct passing 

 through that point of intersection will touch 4s in Arst in other 

 words It coincides with 4s • As however the curves C, and Cs, but 

 not the curves Cr and Ct , neither the curves Cs, Ct have in Arst a 

 contact of the second order we do not find in this way a pair of 

 points satisfying the question. Now it happens three times with two 

 pencils o curves with a common basepoint, between which a pro- 

 jective correspondence has been in such a way arranged that the 



