319 



Theorem 10; Round a parabolical point a finite surrounding exists 

 in the tangent plane containing hyperbolical points only (except of 

 course the original parabolical point itself). 



Let A be parabolical point, a tangent plane and a cuspidal tangent. 

 The points A^, A^ . . . on arch Z?^ converge towards ^. Corresponding 

 tangent planes </,,<f, .... The lines of intersection of «and «i,«, . . 

 are respectively a^, a^ . . . . These lines are tangents in u converging 

 towards a. The points of intersection of a^,a^ . . . and arch AC we 

 indicate l\y C\, C\ .... These points converge towards A. (Fig. 10). 



Let b^,b, be lines respectively situated in «,,«, ... passing 



Fig 10. 



through A^, A^ . . . and perpendicular to «, ,</, .... These lines 

 h^,b^... converge towards b Lci in a. Now let us assume all points 

 A^, A^ . . . were elliptical or parabolical. In Jio case the lines h„ can 

 continue to be cuspidal tangents at A„ in «„ for even if A,, con- 

 tinued parabolical the cuspidal tangents would converge towards a 

 (this follows again from theorem J). Hence for n larger than some 

 finite number the line /;„ has, except A„, another point in common 

 with F^ which counts single. 



bn divides the corresponding plane «„ in two semiplanes. We 

 consider the one that does not contain 6'„. These semiplanes converge 

 towards the top one of those in which b divides a (fig, 10). If A,, 

 is parabolical, then for n large enough the cuspidal branches depart in 

 the semiplane of <'.„ which does not contain 6«, because the branches 

 departing from An cannot switch round 180° in the limiting case 

 (this is shown in a way analogous to that used for theorem 8 where 

 hyperbolical points converged towards a parabolical point). 



Hence for ?i large enough no branch departs from A,, in the 

 semiplane of «„, containing On- But Cu lies on the curve in a,,, 

 and bn carries besides .4,,, only one singly counting point of the curve. 

 But Au is elliptical or parabolical, hence the curve in «„ (with possible 

 exception of A) is connected. From this follows that 6',, must be 



