315 



Theorem G. If a hyperbolical point moves continuously, then 

 the tangents in the tangent plane change continuously also. 



Let the hyperbolical points A^A^.... converge towards the hyper- 

 bolical point A. Corresponding tangent planes «i (i^ . . . h. A sequence 

 of tangents through A^ A^ . . . . in «j a^ . . . . cannot have for limit 

 a line through A in a which is not tangent, for such a line would 

 have, besides A, another point in common with the curve in « and 

 theorem J tells us that the converging lines would carry points of 

 F* having this second point of intersection as limiting point. This, 

 however, contradicts the assumption that the converging lines are 

 tangents at double points converging towards A. 



To prove theorem. 6 it only remains to show that the tangents at 

 the double points in «, «, . . . cannot converge to only one of the 

 two tangents in a. Let a and /; be the tangents in « and let us 

 assume that the tangents in the converging planes «i a^ . . . . have 

 only a as limiting element. For increasing n the tangents in «„ form 

 a diminishing angle tending towards zero. The part of «„ inside 

 this decreasing angle converges to the line a only and considering a 



Fig. 8. 



does not belong entirely to F^ , it is unavoidable that for n > some 

 finite number the part of «n inside the decreasing angle contains 

 that part of the curve which is of the second order and besides 

 this loop of the curve has the point ^1 for sole limiting point (this 

 "loop" can, of course be a projective oval having one or two 

 points in common with the line at infinity). 



Besides the loop, two branches depart from An in the planes «„, 

 belonging to the part of the curve which is of the third order. Let 

 «J a, ... . contain a component sequence «„j a^^ • • • of planes in which 

 these branches depart in the direction of the semitangents converging 



21* 



