115 



2. We now come to the second possibility given on page 111, 

 namely that A is cusp in /i. Again « denotes the plane in which A 

 is double point and b the line of intersection of u and /?. In the 

 proposition of § 3 it was assumed that A is cusp in not more than 

 one plane. Hence if c is a line in a (=[=/;) the point A can never 

 be cusp in any plane through c. Provided c does not coincide with 

 one of the tangents in a either, the reasoning given sub 1 shows 

 that A cannot be double point in any plane through c (except in a). 

 Considering the possibilities given on page 111 it follows that ^ must 

 be ordinary point in every plane through c (except u), with c for 

 tangent. 



Let AF be the cuspidal tangent in ,'i (fig. 4). The line c in a we 

 choose in the same angle of the tangents in A, in which the line 



Fig. 4. 



b is situated. Besides we choose in « a line d through A, not being 

 tangent in A and in /^ a line e, not coinciding with AF or b. The 

 plane through d and e is denoted by é, that through c and J i^ by y. 



The branches meeting at the cusp A in /? arrive from above </ 

 (one on I and the other on III). We consider a sequence of planes 

 y: yi, 72,73 .... turning round AF and converging towards /:{. In each 

 of these planes A is ordinary point with tangent {c^,c,,Cs . . .) situated 

 in «. The branches meeting at A in each of these planes arrive 

 from above « (one on I and the other on III), because the branches 

 in /? arrive from above and none of the lines c^,c^,c^ . . . is separated 

 from ^ by a tangent in .4. 



Each of the lines c„c^,Ct . . . has, except A, another point in common 

 with F\ The distance from A to these points cannot tend towards 

 zero, because if the second point of F^ on b is added, they form 



8* 



