( 769 ) 



By this dispersion lioweAcr new isoljilcd double points aiidisolated 

 bitangcnts are formed, but the higlicr nuinifoid singularities disappear, 

 so tiiat the equation (5) is applicable, provided among ö" and r" the 

 newly generated isolated double points and bitangents are counted. 

 These however can be formed oidy as points of intersection and 

 common tangents of two conjugate imaginary branches. 



Here are three cases to be distinguished with res})ect to the reality 

 of the two points and tangents of the manifold singularity consisting 

 of two conjugate imaginary branches. The case already discussed, that 

 the points and the tangents are both imaginary, does not give rise 

 to a manifold singularity, so it does not come into consideration now. 



Points of contact real, tangents imaginary. From both branches 

 being r(»)ju(/ak' imaginary ensues that the points of contact coincide, 

 l)ut that the tangents ditfei-. If the order of each of the singularities 

 is t, then both branches intersect each other in /'^ coinciding points, 

 which after the dispersion of the singular points cause t'^ double points 

 to be generated. If that dispersion takes place, as w^e keep assuming, 

 in such a way that the equation of the curve remains real, then the 

 singular points become conjugate imaginary, wiiilst the singular 

 tangents remain imaginary. So after the dispersion we get singula- 

 rities which have no influence on the equation of Klein. However 

 we have got another P double points of which t (t — 1) are imaginary 

 and t are isolated. The latter is easy to understand by causing the 

 t coinciding tangents of each of the singularities to diverge a little 

 before the dispersion, by which each of the two singularities changes 

 into a common /-fold point with separated hut slightly differing imagi- 

 nary tangents. The t tangents originating from the one singularity 

 are conjugate to those of the other. With the dispersion the t pair 

 of conjugate imaginary tangents give t isolated points, whilst the 

 remaiiiing double points l)econie imaginary. 



After the dispersion of the singular points by which the number 

 of isolated points has become cf" + t and the number of isolated 

 bitangents has remained in\ariai)le, we find by applying the equation (5) 

 n + ii' -\- 2t" + :£' V, == /• + x' + 2{ö" + + ^' t,. 



For this again we may write : 



n + iï + 2r" + ^' c, = /,- + x' + 2Ö" + ^' ^ , • • (5) 



if we but extend 2't^ to those higher singularities of which the point 

 is real but the tangent imaginary. 



Points of contact imaginary, tangents real. This case is quite 

 correlative to the preceding. Now the points of contact are different, 

 whilst the tangents coincide. Out of this common tangent is formed 



51 



Proceedings Royal Acad. Amsterdam. Vol. VI. 



