( 768 ) 



from this paragraph the manifold higher singularities, point and 

 tangent of conjugate imaginary singularities must both be imaginary. 

 If Ave decompose those singularities into the equivalent ones of 

 Plückkk in the manner indicated l)y Hkill (see note), thus without 

 changing order and class of the curve, then of those PLÜCKEii-singu- 

 larities point as well as tangent are imaginary, as was also the case 

 with the original singularities. So jio PLÜCKER-singularities are formed, 

 which appear in the equation of Klein, so that that equation inva- 

 I'iably holds good for a curve, possessing only higher singidarities 

 of which [)oint and tangent are both imaginary. 



Comprising the results of this paragraph, we thus find for a curve 

 without manifold higher singularities the equation 



H + ,i + 2t" + ^'>\ = k + k' + 2d" -f 2:\ , . . . (5) 

 where the summations must be extended on]y to the real higher dngu- 

 hritles. 



§ 2. Curo'i'.s wltli j'f'td eqitjitioif. and ivith 

 manifold IdcfJier .singidarities. 



If the curve has manifold higher singularities, we can imagine 

 that these are driven asunder in the sepai'ate singularities in such 

 a wa}' by a slight n'ol modification in the equation of the curve 

 retaining order and class of the curve, that its singular points and 

 tangents all differ, but without tlie nature of the separate singularities 

 having undergojie a change. This operation which 1 shall explain 

 more minutely in my dissertation for the case of one manifold 

 singularity only, I call tiie disptn-sion of the manifold singularity ^). 



mation of a higlier singularily •/.*' — /3*' + 2 (S*" — t*") retains the same value, 

 ■which he calls the index of reality of that singularity. Here k*', |3*', I*", t*'' 

 represent the numhers of the real cusps and inflexions and of the isolated double 

 j)oints and bitangenls, generated at the deformation. How large those numbers are 

 separately depends on the manner of deformation. 



This however is not a new result, but an immediate conse(|uence of the equation of 

 Klein if after various deformations we apply it to a curve possessing at first but 

 one higher singularity. 



Brill (1. c. p. 391) says he intends to point out elsewhere that the index of 

 .reality of a singularity (however, this must run: of a real singularity) amounts 

 to J4* — (3*, so according to (4) to t — v. in connection with 



/? + (3' + 2r" + /3*' + 2t*" = ^• + x' + '21" + A*' + 2r " 

 the equation (3) follows immediately from it. However 1 am not aware where 

 Brill gives the promised proof. 



^) So here we leave the one-sided point ot view of the beginning q{ % 1, 



