( J78 ) 



Let K be the de^-elopable formed by the tangents of s. Let / be 

 a common tangent of C^ and S^, touching C^ in R and *' in P. 

 One of the tangents of 6" consecutive to / meets S^ in two real 

 points of s, consecuti\'e to P, as / is supposed to be no ])rincipal 

 tangent of S^. The osculating plane V of C^ in R contains therefore 

 four consecutive points of s, so it is a stationary plane « of .<? in P. 

 Consequently the plane V is also a stationary tangent plane of K 

 along the generating line /. So C^ has in R three consecutive 

 points in common with K and no more. 



The 3?^^ points where C^ meets S^ are cusps ^ oï s^), so they are 

 triple points on the developable K. 



Each of these Zn^ points ^ counts at least for three points of 

 intersection of C" with A^. Each of these points ^ counts for not 

 more than three points of intersection, as we have assumed that C" 

 does not touch S^, and the tangent in ^ to C* does not lie in the 

 triple tangent plane of K in /?, which triple tangent plane coincides 

 with the osculating plane of x in ii, i. e. with the tangent plane of 

 aS; in iJ. 



The curve C^' meets /v only in the 4 w^ — • '' ^ — V^ points R 

 and in the ^n^ points /?, as every tangent to s lies in an osculating 

 plane of C', and through a point of 6" no plane can pass osculating 

 6'^ still elsewiiere. The order of K or the rank of .>• is 

 r — 4m, + 3w, — 26 — 3/ '). 



So the number of points of intersection of C^ and K is 

 3(4w, + 3», - 2d— 3x). 



As the only points of intersection of 6'^ and A' are the points 

 R and /i counted three times, we find the relation 



3(4m, f ^n, - 2ff - 3x) = 3 X 3^^, + 3(4;//., - .'-^ - ?/•/) 



from which ensues 



X =z 2, 2/ = 3, 



or in words: 



If the developable D* and an arbitrary surface S., have an ordinary 

 contact, two consecutive generating lines of D^ touch S^. 



If the developable D" and an ordinary surface S^ have a stationary 

 contact, three consecutive generating lines of D* touch ^S',. 



These theorems hold good too in the case that the developable is 

 a cone '). 



1) Versluys, Mém. de Liège. 3me série, T. VI, 1905. Sur les nombres Plücké- 

 riens etc. 



2) Versluys, These Proceedings May 27, 1905. 



3) Versluys, These Proceedings May 27, 1905. 



