316 



towards the semiline AB. This implies no restriction, because with 

 respect to the curve in « both possibilities indicated in fig. 8 are 

 considered. 



Let c be a line through ^ in « inside the angle EAB. This line 

 lias, besides A, another point in common with the curve in a. Let 

 c„i c„3 .... be a secpience of lines through A^^ Ay,^ . . . , respectively 

 situated in «„^ «„, .... and converging towards c. The jioints Pn^ P,,^ . . . 

 on these lines are supposed to have P for limiting point. Let a,,, «„, . . . 

 be a sequence of tangents in «„j <t„^ . . . and lastly we assume that 

 the points i?„j B„^ ... on these lines converge towards B. (see fig. 8). 



In f(„, a branch departs from An^, between A,,^ B,,^ and .4,,, P„i, 

 in «„, a branch departs from A,,^ between A,,^ B,,^ and A,,^ P,,^ etc. 

 These branches cannot cross the tangents A,,^ B^^ etc. again hence 

 in order that in « no branch leaves A between AB and AP it is 

 necessary tiiat the branches in the converging planes cross A,, P,, 

 in points converging towards A,,- According to theorem^, how ever, 

 the lines c„ (of which A» P„ forms part) end up by carrying 

 points of F^ converging towards R and because A„ counts, double 

 lines would exist having four points in common with F^. 



The tangents at a double point divide their plane in two parts, 

 respectively containing the loop of the curve and the part of the 

 third order. 



Theorem 7 : //' a hyperbolical point moves continnously , hence the 

 tangent plane loith tangents also changes continuously , then the parts 

 of the tangent planes, containing the pieces of the tJiird order, merge 

 in each other and it follows that the same holds for the parts con- 

 taining the loops. Besides the loop cannot switch round 180°. 



A^ A^ . . . . converge towards A (all hyperbolical). Tangent planes 

 ffj «2 . . . . «. The tangents in «„ form four angles round An, succes- 

 sively I„, II,„ III„ and IV„. These converge respectively towards I, \l, 

 III and IV in a. Suppose for every n the principal branch (that is 

 the part of the third order) in «„ lies in I„ -|- III„. This means 

 that in «„ branches depart from A to both sides inside these angles, 

 which are connected via the line at infinity. But then it is unavoidable 

 that in the limiting plane a a branch departs in I and another in 

 III, hence this part of a again contains the principal branch. 



Suppose for every n the loop in «„ departs in II„. In order that 

 A be isolated in « in the angle II it is necessary that these loops 

 contract towards A. Hence, in the end thev cannot reach the line 



