( 55 ) 



pass two consecutive tangents of .v. Whence already 2 n tangents 

 of s meeting /. 



Tangents of s, being at the same time generating lines of /{'^ pass 

 through the vertex T oï K^ and, being tangents of 6% are also tangents 

 of S^, and therefore situated on the tangent cone Iv of S^, having T 

 for its vertex. Conversely every common generator of tlie two cones 

 K"^ and K is a generator of K^ having with >Sj, thus also with s, 

 two coinciding points in common. A right line having with s two 

 coinciding points in common is either a tangent of s or it passes 

 through a double point of s. So tiie common generators of the 

 cones lO and A' are either tangents of .s" or they pass through double 

 points of s. The order of the tangent cone K, being equal to the 

 class m of >S\, the number of common generators is 1m. The num- 

 ber of tangents of s meeting / in the vertex J' will be %n, diminished 

 by a number still to be determined for the common generators 

 passing through a double point of s. 



If K^ has in a point ö an ordinary contact with S^ the common 

 tangent plane jt in cf is a tangent plane of S^ passing through T. 

 So .T is also a tangent plane to the cone A" along the line Té. 

 So the two cones A^ and K have along the common generator 

 Té a common tangent plane. The line Té must therefore count 

 for two common generators of the cones K^ and K. A point d is 

 a node of s and with the exception of very particular cases the two 

 tangents of 0^ in d will not coincide with Té. So for every point 

 d the number of tangents of s passing through 7' must be diminished 

 by two. 



The following example proves that for every point / in which 

 >Si and K^ have a stationary contact, the number of generators of 

 K"^ touching s must be diminished by three. Let /S'l also be a 

 quadratic surface and let the curve of intersection s be a not degene- 

 rated biquadratic curve R"^ with a cusp /. Then the line 7/ counts 

 already at least for two common generators of the cones K'^ and K 

 and is again not a tangent in x to s or R^. If now^ Tyi were to 

 count only for two common generators the cones lO and K would 

 have two more generators in common. These latter two cannot be 

 two consecutive generators, for in that case R'^ would ha\'e two 

 double points and so it would have to break up. Now it is easy to 

 see that these two remaining generators are tangents to R^ or s at 

 points for whicii the osculating plane is a stationary plane. So R^ 

 would have to possess two stationary planes « whilst a R* with cusp 

 possesses but one stationary plane a ^). The right line 7/ must 



1) E. Pascal, loc. cit. p. 363. 



