310 



Now we obtain a contradiclioii if we can show that none of these 

 semiplanes in which the convex arch departs above {below) the 

 tangent in A can be limiting element of a sequence of semiplanes 

 in which tiie convex arch departs behiv {above) the tangent. 



Let the semiplanes «j, «,, a, . . . . converge towards «. Let the 

 corresponding semitangents be b^, b^, b^ . . . . b. In «j, «,, (^^^ . . . . the 

 convex arches are assumed to depart below b^, b^, &,...., and in u 

 above b. The lines b^, b^,b^ . . . . have at least one limiting line b' 

 through A in a. There are three a priori possibilities: b' can be 

 situated above b, below b or can coincide with b. 



First case. Let b" be a semiline through A between b and b', 

 having a second point B in common with the convex arch departing 



from A in (t (fig. 7). Let i?" be the plane 

 through b" ± a. Let the lines of intersection 

 of t?" and «1, ((^, «,.... be respectively 

 ^i". ^,". ^s" • • • • From ^1, 6, . . . we choose 

 a component sequence :6„i, è,,, ^„,. .. . having 

 6' for sole limiting element. Corresponding 

 sequences are «„,, «„,, «„,, .... and 6",,,, 

 '^"»!2, b'\,t . . . For «, large enough the semi- 

 lines ^„1 bn^ . . . (converging towards b') are 

 ^^ë- '^' respectively situated above b"„^, bn^,bm . . . . 



(converging towards b"). 



In «,n a branch departs from A between 6», and b''„^, in «„, a 

 branch departs from A between bn^ and ^"„, etc. None of these branches 

 can cross bn^ b„, b„z... (respectively) because these lines as tangents 

 at the point of inflexion A have no other points in common with 

 F*. Hence in order that in the limiting semiplane a no branch 

 departs from A between b' and b" it is unavoidable that in the 

 converging planes the branches cross b",i^ b„'\ b",,^ ... at points con- 

 verging towards A. But all the lines b",,^ b"„^ . . . are situated in 

 one plane ^", hence in this plane b" is tangent in A. Then however, 

 it is impossible that A is point of inflexion in ,i", because b" has, 

 except A, another point B in common with F*. 



Second case. Possibly the line a has, below A, one or two other 

 points in common with F^. Let C be the nearest. Again b^, b,,^ ■ . ■ 

 represents a sequence of semilines having // for sole limiting element. 

 In «„J, «„3 .... curves depart from .4 below b^^ etc. and to the right 

 hand side of a. These curves cannot recross b,,^ etc. because these lines 

 have only A in common with F^, hence in the lower angle between 

 a and b^ etc. these curves connect A either with the line at infinity 



