( 56 ) 



therefore count for three common generators of K^ and K. The 

 number of tangents of s meeting the line /, thus the rank of s is 

 consequently 



r= 2?i + 2m — 2d — 3/ . 



6. The reciprocal polar figure s' of the curve of intersection s of 

 K^ and S^ is a developable circumscribed to a conic c* and to a 

 surface S' of order m and of class n, whilst the conic c^ touches 

 Ö times the surface >S'' and osculates it x times. If we take for the 

 conic c^ the imaginar^^ circle at infinity the developable s' becomes 

 the developable focal surface of S'. The rank of s is the same as 

 that of s. So we find the theorem: 



The rank of the focal developable of a surface of order m and of 

 class n touching the imaginary circle at infijiity ö times and oscidating 

 it X times is 



r = 2m-\- 2n — 2(f — 3x. 



If aS\ is a developable the point of contact of a common tangent 

 plane that is an ordinary plane of S^ is> always a node of s ^). 

 The developables K^ and S-^ will only then have a stationary 

 contact in a point Xj when the common tangent plane is a stationary 

 plane a of ^S\. The line 7x counts thus for four lines of intersection 

 of the cone K^ with the tangent cone K which breaks up into 

 m planes. It is easy to see that now the line 7x is at the same 

 time tangent to s at the special cusp x wdiich is a singularity of 

 order two, of rank unitj' and of class three ^). So a stationary contact 

 X gives rise to four lines of intersection of K^ ^vith K of which 

 only one is an ordinary tangent of s lying on /v^^ Each point x now 

 also diminishes the rank of ^^ b}^ three. The reciprocal polar figure 

 of Si is a curve S' of order m and of class ??,. Each common tangent 

 plane of K^ and S^ is transformed in a common point of c^ and 

 S'. If the common plane is a stationary plane « of S^ the common 

 point is a cusp on the curve S'. So we find the theorem: 



The rank of the focal developable of a plane curve or a twisted 

 curve of the degree m and of the class n and of ivhich ö ordinary 

 points and x cusps lie on the imaginary circle at infinity is 



r — 2m -f 2/i — 2(f — 3x. 



7. If &\ and 8\ are the reciprocal polar figures of the surfaces 

 aSi and S^, then 8\ and 8\ are respectively of the degi'ee m^ and 

 m.^ and of the class n^ and n^. 



1) Versluys, Mémoires de Liège, 3me série t. VI. loc. cit. 



2) Halphen, Bull, de la Soc. Mat. de France, t. VI, p. 10. 



