305 



ever}' plane /?„ the line a„ is ordinary tangent at A. It is easy to 

 show that in the vicinity of A the curves in all these planes lie 

 on the same side of «. However this result is not wanted : we 

 merely take a component sequence of planes in which the curves 

 depart from A on the same side of «, let us say above «. 



The point A divides the cuspidal tangent a in two semilines, 

 let a' be the one departing from A in the same direction as the 

 cuspidal branches and a" the other. The corresponding semilines on 

 the converging lines we denote by a/,a^' ... and ai",a," 



In every plane [i,, a branch departs from A above « in the 

 direction of cin". The reasoning used for the examination of a section 

 in a plane through a tangent at a double point shows here that in 

 f? the limiting branch departs from A above a in the direction of 

 a". The line a has only A in common with F^ and considering A 

 cannot be double point or cusp in /?, the only remaining possibility 

 is that A is point of inflexion in ^ with a for tangent. 



This completes the proof that a is tangent plane. 



§ 5. If A is cusp in two dijf event planes, then A is exceptional 

 point. 



In § 1 it was shown if A is isolated in a plane ((, then a is 

 tangent plane or A is exceptional point. In case n is tangent plane 

 it was found that A is ordinary point in every plane except ((. 

 Hence when A is known to be cusp in some planes, and if we 

 want to show that A is exceptional point, then it suffices to prove the 

 existence of a plane in which A is isolated. 



Point A is cusp in two different planes. We consider two assump- 

 tions: that the two cuspidal tangents coincide or do not coincide. 



First assumption: A is cusp in the planes a and ^? and the line 

 of intersection a of these planes is the common cuspidal tangent. 

 Let y be a plane through A not containing the line a. 



In a the point A counts double on the line of intersection of 

 «and/, hence, according to the theorem of page 117 —118, ^ also counts 

 double on that line of intersection in y. The same holds for the 

 line of intei'section of /? and y. Hence in y two different lines exist 

 on which A counts double, and from this follows that A is in y 

 either cusp, double point or isolated point. 



If A were cusp in y, then A would be cusp in two planes « and y 

 and the cuspidal tangents would not coincide. This case shall be 

 dealt with later on, when the second possibility is assumed. 



Hence to show that A is isolated in y it only remains to prove 

 that A cannot be double point in y. 



