( 766 ) 



whether we do or do not inolude entirely real or entirely imaginary 

 non-singular elements among Ihe 2i^'-signs. 



The equation (2) holds ijood for curves irlth inuufinart/ erjuutioii 

 as ivell as for curves with real equation. 



Now follo\y the eliief points of the deduction of the equation 

 discussed. This will be giyen moi'C minutely iu my dissertation ^) 

 still to appear. 



To this eiid we shall treat in §1, 2 and 3 the relation (2) for 

 curyes with real equation, taking that of Ki-ein as our starting 

 point. In § 4 \ye shall indicate it for curyes with imaginary equation 

 too, and in § 5 we shall transform it to other forms. 



§ J. Curves with real ('(/uadou and u:ith no odu'r manifold 

 sin<fuhfrities than (hnihli' points aiul hitainjcnts. 



For the present we shall take a one-sided point of yie\y where a 

 cnrye is regarded as a locus of points. 



If the curye has higher nnifold singularities we dissolve them. 

 This means that we liring about such a small real modilication in 

 the equation in j)oint coordinates \\\{\\ ])reseryation of the order, 

 that the higher singularities disa[)pear without Pi,ü(KKK-point-singn- 

 larities (cus])S and double ])oints) taking their i)lace (but of course 

 intlexions and bitangents). After this dissolution, where we assumed 

 the Pluck r-R-singularities already present to be remaining, we apply 

 the equation of Klein. 



To this end Aye must consider how many isolated bitangents and 

 how many I'eal intlexions appear in the dissolution of a higher 

 singularity. In two ways isolated l)itangents can be formed, iiatnely 

 l^t ill the dissolution of a rad siiu/ularifi/, i. e. a singularity 

 whose correspontlijig singular Itranch is real, 2'"^^ in the dissolution 

 of two conjugate imaginary singularities; here })oint as well as 

 tangent must i)e imaginary, as otherwise we should be treating a 

 manifold singularity, which we exclude from this paragraph. Of 

 course real inflexions can arise only from the dissolution of real 

 singularities. 



By dissolving the singularity the class of the curve undergoes 

 an increase d. Here (/ represents the reduction, of class of the singu- 

 larity (called l)y Smith, I.e., }). 155 the discrimiinintal inde.r) i.e. the 



1) Over den iuvloed van hoogere singulariteiten op aanrakingsproblemen van 

 vlakke algebraïsche krommen. (On the inlluence of higher singularities on problems 

 of contact of plane algebraic curves.) 



