320 



connected with the line at intinitj in the corresponding A'ö///fplane of «». 

 Passing on to the limit the semiplane of «„ containing C'„ con- 

 verges towards the top semiplane of « (fig. 10) and 6',, converges 

 towards A. Bnt then it is unavoidable that a branch departs from 

 A in the top semiplane of « (including line h). Hence a contradic- 

 tion is obtained and it has been shown that the points A^, A^ . . . 

 end up by being hyperbolical. 



Theorem ll; If a surface F* contains no straight line, it 

 cannot exist. 



JuEL has shown ^) that a non-degenerated elementary curve of the 

 third order contains one and only one point of inflexion if that curve 

 has a double point or cusp and that otherwise the curve has three 

 and onl}' three points of inflexion. We consider an arbitrary plane 

 section of F^. If this section contains no straight line it has at 

 least one point of inflexion. This point can be hyperbolical or para- 

 bolical. In the latter case we can, according to theorem 10, find 

 hyperbolical points in the corresponding tangent plane. Hence in any 

 case a hyperbolical point of jP" can be found. Let this be A with a 

 for tangent plane. We do not consider the loop of the curve in «, 

 but only the principal branch. This branch has, according to Juel, 

 one and only one point of inflexion B. We consider two points 

 A' and A'\ departing from A in opposite directions along the principal 

 branch and meeting again at the point of inflexion B, after moving 

 continuously along the curve. Theorem 9 shows that the loops of 

 A' and A" , at first cross a and besides we found that at first they 

 trail behind. Now in this state of affairs no change is possible before 

 A' and A" meet again at B: The tangents at a double point 

 change continuously {theorem 6), hence only at a point of inflexion 

 can they pass through a. Besides the loop cannot switch round 

 180° or change into a principal branch [theorem 7). Lastly it is 

 impossible that on the way from A to B the point A' (or A") 

 changes its character, for this can only happen via a first parabolical 

 point. Then, however, the angle between the tangents, inside which 

 the loop is situated, would tend towards zero, and considering the 

 loop crosses « all the time, the limiting position of the tangents 

 would be situated in « and this would mean a cusp or point of 

 inflexion in «, because the above mentioned limiting line coincides 

 with the cuspidal tangent {theorem 8). Hence A' and A" arrive at 

 the point of inflexion B both hyperbolical with the loops crossing 

 « and trailing behind. 



^) Proc. Danish Acad, loc cit. § 5. 



