318 



departs from A along the principal branch in a, then at first the 

 loops in the corresponding tangent planes will cross plane a. 



Let lis consider a sequence of points on arch CM (fig. 9) converging 

 towards .4. Provided we start close enough to A all these points 

 are hyperbolical {theorem 5). We assume that the loops do y^c/ cross 

 plane « and we select a component sequence of points such that the 

 corresponding loops all arrive from the same side of a for instance 

 from above. Let this sequence of points be A^ A^ . . . . Tangent 

 planes «j, «,, 



Fig. 9. 



The tangent to .16' in A,^ intersects arch >4i^ in Z»,i. For increasing 

 n, Bn converges towards A. Plane «„ contains line J,, 5,.. Let J„ Ai 

 be the semitangent in «„ which converges towards AD. Let us con- 

 sider in iin that part of the plane inside angle A, An B,.- ^or 

 increasing n no principal branch in «„ can continue to depart inside 

 this angle, because the principal branches depart from A,^ in direc- 

 tions converging towards A E and AF. B,,, however, belongs to the 

 principal branch, as we assumed that the loop does not cross «. 

 This principal branch which crosses An B» at B„, cannot connect Bn 

 with An inside angle Bn An Dn, but on the sides of this angle B» is 

 the only point of 7^^', because An counts double on An B„ and triple 

 on .4,, Dn, hence it is unavoidable that inside angle B„ An Dn this 

 branch connects /i„ \vith the line at infinity. Angle Bn An Dn, how- 

 ever, converges towards the semiline AD and Bn towards A. It 

 follows that line AD would belong entirely to i^^ : a contradiction. 

 This completes the proof of theorem 9. 



Remark: When a point departs from A along the principal branch 

 in « the tangent can be divided in a front half and a back half. 

 The preceding proof shows that at first the loop departs inside that 

 angle of the tangents, in which the back half of the tangent in « 

 is situated. We shall indicate this briefly by saying that at first the 

 loop trails behind. 



