( 767 ) 



mnltiplicitv of the singular point as point of intersection of the curve 

 with the lirst polar of an arbitrarv point. 



Influence of a rkat. singularity. Suppose when dissolving a 

 real higher sinciularitv with a reduction of class r/^, we arrive nt ^\ 

 real inflexions and t'\ isolated bitangents. The class then becomes 

 /■_!_(/ so that ensues from the equation of Klein, if for simplicity's 

 sake Ave think the cu)-ve to possess but one higher singularity: 

 M + ^' + 2t" 4- ^\ + -^r", = k + '/, + h' + 20". 



What the value is of ,^\ and of t'\ separately, depends upon the 

 manner of dissolution. If however Ave a[)ply the above equation to 

 curves formed in different manners of dissolution, we find that 

 ,:?\-f 2r'\ has always the same value, called by me the reduction of 

 reaUtij of the singularity. In my dissertation I shall deduce out of a 

 definite manner of dissolution for that reduction of reality the 

 value d^-\-i\ — t-^, which causes the latter equation to become 



n + .i' + 2t" + V, = k + k' + 2d" -f- ^ ^) . . . . (3) 



Influence of two conjugate lmaginaky sin(;ularities. As we excluded 



1) This equation agrees with the uidex of rmUty given by A. Brill (Uebcr 

 Singularitiiten ebener algebraischer Curven und eine neue Curvenspecies. Math. 

 Ann., Bd. 16 (1880), p. 848, spec. p. 391) based upon the decomposition of the 

 higher singularity in PLÜcKER-singularilies. A. Cayley (On the Higher Singularities 

 of a Plane Curve. Quart. Jouni. of Math., Vol.! (1866), p. ^2\% CoUected Math, 

 papers, Vol. 5, p. 520) has namely shown, although in a not entirely satisfactory 

 way, that the PLÜCKER-equations as well as the equation of deficiency keep holding 

 good for curves with higher singularities, if we regard such a singularity as equi- 

 valent to X* cusps, |3* inflexions, 5* double points and t* bitangents. For k* and ^* 

 Cayley gives 



^* = t-[ , ^* = v-l , (4) 



and he indicates how 5* and t* can be deduced from the PuiSEUx-developments 

 in point and line coordinates. 



Later on fuller proofs for the results of Cay'ley have been furnished, among wliicli 

 that of Stephen Smith excels for its simplicity and rigorousness (1. c, p. 153— 16:2), 

 based upon the line of thoughts of Cayley. For the CAYLEY-numbers of equivalence 

 Smith introduces (1. c , p. 161) the names cuspidal index, uiflexional i)id(\c,)iodal 

 index and Ijitangetitial index and among them he finds a simple relation (1. c, p. 166). 



Brill has shown that this CAYLEY-equivalence does not only completely satisfy the 

 PLÜCKER-equations and the equation of deficiency, but tliat if is possible to deform 

 the curve retaining order, class and deficiency in such a way that the higher singu- 

 larities are decomposed into the equivalent ones of Plücker. Brill calls tliis opera- 

 tion a deformation of tiie singularity (I.e. p. 361). Already Cavlev (On the Cusp 

 of the second kind or Nodecnsp. Qnart. .Journ. of Math., Vol. 6 (1864), p. 74, 

 Collected Math, papers, Vol. 5, p. -265) gives for the case of a ramphoid cusp an example 

 of a such like deformation although he does not emphatically draw the attention 

 to the fact tliat class and deficiency remain unaltered. 



hi an elegant way Biull indicates further algebraically, that for every real defor- 



