( 111 ) 



If we investigate vvhicli ralional curves of (lie theorem II coiitii- 

 bnte to 2£ {il' — A\) then we fiiul 1^^ those rational cui'ves wfiieli 

 pass through an origin (lying outside the hasepoints of the pencil) ofa 

 superUncar branch (denomination of Oaylky for a l)ranch with order 

 more than tuui) of the tixed curve, 2'"' those rational ciirxcs louchiiig 

 the fixed curve outside the hasepoints, 3'° those rational curves 

 where two or more of the movable i)oints of intersection have 

 appi'oached one of the hasepoints along the same branch of the fixed 

 curve. In the main theorem 11 comes to (he determination of tlie 

 number of rational curves of a pencil toiichijig a gixen curve and 

 the change which this number luidergoes on account of higher singu- 

 larities of the given cui-ve and the particular situation of the hasepoints 

 with respect to that curve. Here however it is diflicult to imagine 

 how it would cause a difference whether we ar(> working willi a |»eucil 

 of rational curves or with an arbitrary {)eucil of curves; for in both 

 cases the coefticients of the e(|uation of the iuo\al>le curve satisfv 

 some linear conditions amounting to one less than is necessary for 

 the defiuiliou of the moxable curve '). 



Let us explain the preceding by an example. Suppose the numl)er of 

 cubic curves through eight [)oints touching a given curve were re(|uire<l, 

 supjiose further that the given curve has a singular point N with a 

 singular tangent / and Ihat the ^ '., of the pencil Ihroiigh N also 

 touches /. Point /S will then absorl) a cerlaiji niind)ei- of |)oinls of 

 cojitact pro|)er with ciu-ves of the pencil, and tliis luiinber will depend 

 on the Jiatui-e (»f the singular point *S' and on the oi-der of contact 

 of 6'., with the given curve, but in no wise on the fact 

 wdiether of the basepoints three have coincided somewhere otitside S, 

 either in such a way I hal in one of the basepoints tangejd and 

 curvature are given, oi- in such a way that the coinciding basepoints 

 form a triangle with finite angles, in which case the condition of the 

 passage through the three basepoints includes the curve having a 

 double point in a given point and being thus rational. 



Led by the above considerations 1 thijd^ I may slate the following 

 theorem : 



Theorem 111. //' an tihichiuiic i>l<in<' cuiti' is nitri-sccti'd hji a jwiicil of 

 curves in, ii' inovdlilc poii/ts ilistfiiinlnni lln'niselvi's for the ili ffrn'iit 

 ciwves of the jjeiicii orer .V,. X.^, . . . hrmirlws of tin' fi.vcd curve, 

 then 1 — n.' + ^ -S" (//' — A',), (2i^ (// .V,) fu/,-cn i>r,'r nil fin' curves 



1) Of coufso il, would l»e dirfcrciil il' llic movable I'lirve had lo be rulioiial williuut 

 the singular |)oiiits laliiciiiü,' llu' ,ü,<'iuis to zero being given ; if thus e.g. tlie question 

 were of (■ii])ic curves tlirougli seven uiveti |)oinls, and furnislied with a double point 

 not given in position. 



