( 557 ) 



3. To determine the order of the surface of contact of a hootold 

 infinite system of twisted curves and a singly infinite system or' 

 surfaces. 



To this end we shall first suppose the two systems to be arbitrary. 



To determine the order of the surface of* contact we count its 

 points of intersection with an arbitrary right line /. To this end we 

 consider the envelope E x of the oc' 2 tangential planes of the curves 

 of the system in their points of intersection with / and Hie envelope 

 E, of the oo 1 tangential planes of the surfaces of the system in their 

 points of intersection with /. 



The common tangential planes not passing through / of both 

 envelopes indicate by means of their points of intersection with / 

 the points of intersection of' / with the surface of contact. 



In order to find the class of the envelope E\ (formed by the 

 tangential planes of a regains with / as directrix) we determine 

 the class of the cone enveloped by the tangential planes passing- 

 through an arbitrary point Q of /. If the system of curves is such 

 that <p curves pass through an arbitrary point and if' curves touch 

 a given plane in a point of a given right line, the tangential planes 

 of E l through Q envelope the <p tangents in Q of the curves of the 

 system through Q, and the line / counting ip times; for each plane 

 through / is to be regarded if? times as tangential plane, there being 

 if> curves of the system cutting / and having a tangent situated in 

 this plane. The envelope E x is thus of class tp -\- if? and has I as 

 xy-fold line l ). 



To find the class of the envelope E % we determine the number of 

 its tangential planes through an arbitrary point Q of /. If now the 

 system has (i surfaces through a given point and r surfaces touching 

 a given right line, the tangential planes of the envelope passing 

 through Q are the tangential planes in Q to the (i surfaces passing 

 through Q and the tangential planes of the v surfaces touching /. «So 

 the envelope E 2 is of class (i-\-v with v tangential planes through I. 



Hence both envelopes have (</> f if') {[t \-r) common tangential planes. 

 Each of the v tangential planes of E t passing 'through / is however 

 a if-- fold tangential plane of E, and so it counts for if' common 

 tangential planes. So for the number of common tangential planes 

 not passing through /, thus the number of points of intersection of/ 

 with the surface of contact we find : 



(<P + *f') (ft + ») — 'f'i> = *pv -f if7* h W i 

 therefore : 



l ) The regulus as locus of points has however line I as £-fold line. 



