317 



at infinity and it follows that in a the point A must be isolated 

 also in the angle IV. It follows that the oval cannot switch round 180°. 



Theorem 8: //' a sequence of hyperhoUcal points converges 

 towards a parabolical point then both sets of tangents of the hi/per- 

 bolical points converge exclusively toivards the cuspidal tangent of the 

 parabolical point. Besides the direction in luhich the principal branches 

 (that is the parts of the third order) depart, cannot switch round 180°. 



Let the hyperbolical points A^,A^... converge towards the parabolical 

 point A. Tangent planes «,, «, . . . a. A line through ^ in « which 

 is not cuspidal tangent has, besides ^, an ordinary point ^ in common 

 with the curve in a. Theorem 1 shows that such a line can never 

 be limit of tangents at double points converging towards A. This 

 proves the first part of theorem 8. In the same way as when proving 

 theorem 6 we can show again thai the loop in the converging planes 

 lies inside the diminishing angle of the tangents and converges 

 towards A. 



In the case of theorem 7 it was shown that the loop cannot switch 

 round 180°, and it followed that the principal branch cannot change 

 its direction discontinuously either. Here, however, the loop has dis- 

 appeared in the limiting plane a and a new proof is required that 

 the principal branch cannot change its direction in discontinuous 

 fashion. 



Let AC be the cuspidiil semitsingeni in a departing in the same/ 

 direction as the cuspidal branches and AB the other half. Suppose 

 in the converging planes the principal branch departs from A„ in 

 the direction of the semitangents converging towards ^J5, but in « in 

 the direction AC. Let 6 be a line through A in « (=i= BC) and 

 b^, b, . . . a sequence of lines converging towards b respect, situated 

 in «!,«,... and passing through A^, A^ . ■ . 



The principal branches depart in «« from A» to both sides inside 

 the increasing angles of the tangents. These branches do not cross 

 the tangents again, but each goes in its own angle to the line at 

 infinity. Besides they depart in directions converging towards AB. 

 In order that in « no branches depart from A to that side of b on 

 which AB is situated it is necessary that both branches in a,t (for 

 71 large enough) cross the lines bn (converging towards b) on both 

 sides of An at points converging towards A. However, on every 

 line bn the point An counts double, hence lines would have been 

 constructed having four points in common with F\ 



llieorem 9 : // .1 is double point in plane u and if a point 



