( 183 ) 



point P, the tangent in P to 6'' coincides with the principal tangent 

 in P of S, whilst the osculating- plane of C'' in P coincides with the 

 tangent plane of S in P. From the expressions [D) for a^,a^ and 

 rt„ follows that the discriminant (/s) is divisible by t\ so that C" and 

 S have now four common tangents in common in I he j)oinl P. 



If h =^ b z=z p z= then C touches S still in a parabolic point ; 

 the only difference to the preceding case is that 6''' Jio longer 

 touches the prijicipal tangent. From the equations (/)) and {E) ensues 

 that 6'^ and S possess only three common tangents. 



If h ==z b = and p ^ 0, then P is a parabolic point for which the 



principal tangent does not coincide with the tangent to 6". From 

 (D) and (E) ensues now readily tiiat the A'-axis counts but for two 

 common tangents of C^ and vS. 



•o^ 



§ 7. When the A'-axis coincides with one of the principal tangents 

 of S in P then the axes cannot be taken in such away that A = 0; 

 but we have now a = 0. The terms of the lowest order in t in the 

 coefticients a^, a-^, a^ {D) are now respectively of degree 2, 1. 'J. 

 So the discriminant {E) is onl}^ divisible by f, so that now the A"- 

 axis counts for two common tangents of C^ and S. The A^-axis itself 

 has now with S in P three consecutive points in common, so it 

 counts already for two common tangents. A tangent of C^ following 

 the A'-axis does not touch S any more. 



The term of the second degree in t of the discriminant {E) has 

 now for coefficient IQCh^p^, where 6' is a constant. So the discrimi- 

 nant has three roots ^ =: 0, when h^O or p = 0. The case A = 

 is just the one treated in § 6. 



If J) = a = then C^ touches in P one of the principal tangents 

 of S in P, whilst the osculating plane of C'^ in P still coincides 

 with the tangent plane of S in P. Out of the expressions (D) for 

 cIq, a^ and a^ it is evident that these coefticients are respectively^ 

 divisible by f, f and t. So the discriminant {E) is divisible by /^ or 

 it has four roots t = 0. The A'-axis counts thus for four common 

 tangents of C^ and S. By substitution of x = t, y = f , 2^ = t^ in 

 the equation {B) of the surface S we find that C" and >S' now have 

 in the origin three consecutive points in common. 



^ 8. Let 6\ now be an arbitrary twisted curve and D^ the 

 developable formed by its tangents and let D^ touch the arbitrary 

 surface S in P. Let / be the generating line of Z), touching S in 

 P and let 11 be the point, in which it touches 6\. Let V be the 



