108 



two planes passing through c and through a' and cd' respectively 

 show oi'dinarj points in A. Hence (using the results obtained above)' 

 every plane through c shows an ordinary point in A and all the 

 tangents are situated in a. 



But c is an arbitrary line through A only subjected to the con- 

 dition not to lie in a or /?, so it follows that in every plane, except 

 (( and {i, A is ordinary point with tangent in u. Besides in ^ A was 

 assumed to be ordinary point and the tangent was found to lie in a, 

 hence the only remaining exception is u in which plane A is isolated 

 and which has now been proved to answer our definition of tangent 

 plane. 



We now assume the second possibility given above: 



The point .4 is isolated in a and cusp in /l Let 6 be the cuspidal 

 tangejit. In no plane through h can A t)e ordinary point, for iflliis 

 were the case, it might be shown in the same way as above that 

 A cannot be cusp in /■?. Also in no plane through b can A be 

 isolated because b has only the point A in common with F'^. Taking 

 into consideration that above u there is a finite vicinity of A 

 containing no points of F^, the only remaining possibility is that ^ is 

 cusp in every plane through b. b must be cuspidal tangent in every 

 one of these planes because b has only the point A in common 

 with i^'. Now a cnsp counts double as point of intersection on any 

 line except the tangent, hence every line through A {=\=b) has one 

 and only one other point in common with F^, because in the plane 

 through that line and b the point A is cusp with b for tangent. 

 Thus in a plane through ^l which does not contain 6, every line 

 through A has one and only one other point in common with F^, 

 hence A is isolated in every plane which does not contain b. Thus 

 it has been shown that A is exceptional point. 



Before proceeding further we shal] just rehearse what has been 

 done in § 1 : 



A was assumed to be isolated in plane <<. Then a plane /? was 

 constructed in which A was not isolated. From the assumed isolation 

 in a it followed that only two things were possible, namely that A is 

 ordinary point in /? with tangent in a or that A is cusp in ^. 

 Assuming the first possibility we proved that a must be tangent 

 plane, while the second assumption lead to the conclusion that .1 

 is exceptional point. 



§ 2. Only one ed'ceptional point is possible. 



Suppose there could be two: A and B. In a plane through A 

 and B there are a priori four possibilities: 



