( 558 ) 



The surface of contact of a system (<p, »f\ oj 3o s twisted curves' 1 ) 

 ami a system (fi, v) of x 1 surfaces ') is of order tpv -[- i|*;t ~\- g>p *). 



4. To determine the order of the surface of contact 4 ) of the system^ 



fij , v x ) , (ft, , r 2 ) and (jit, , r 3 ) each of x 1 surfaces, we regard the 

 system (<p , if') of the curves of intersection of the systems (px 1 , »J and 

 (f* a , r a ). Of these curves of intersection fop, pass through a given point, 

 so <p:=n 1 fi a . Tlie if? points, where the curves of intersection touch a 

 given plane in a point of a given right line, are the points of inter- 

 section of that given line with the curve of contact of the systems 

 (/ij , v x ) '") and (ft, , v, pf plane curves, according to which the given 

 plane intersects the systems of surfaces (fi l ,v l ) and (f«, , v s ). This 

 curve of contact is of order p 1 v i -f n i v 1 -f f<iM 3 , thus : 



tp =f* t p a + fi^i', 4 fijfv 



The surface of contact to be found is thus the surface of contact 

 of a system (f*if* s » f ï^a ~r~ 1*% V \ ~\~ Mifd °f x * twisted curves and a 

 system (fi 8 , v,) of x 1 surfaces, so that we find: 



The surface of contact of three systems (n l , vj, (ft, , i\) and (ju, , v 8 ) 

 o/ x 1 surf tecs is of order 



!>J' S v i + MafV, + fife's + BPiPiPf 



If the three systems are the pencils (/%,), (Ft) and (F lt ) we have 



Mi =M 3 = f*. = !, 

 r 1= 2(6 -1) , r, = 2(«-1) , », = 2(«-l). 

 So we find : 



77/« surface of contact F stu of the three pencils of surfaces (F s ), 

 F t ) and (F u ) is of order 



') System with f curves through a given point and J/ curves cutting a given 

 line and touching in the point of intersection a given plane through that line. 



~) System with m surfaces through a given point and v surfaces touching a 

 given right line. 



3 ) This result is also immediately deducible from the Schubert formula 



.rp-2 = p's . g + p'g'c . p~g e + p i% . p~g e 



(Kalkül der abzahlenden Geometrie, formula 13, page 292) for the number of 

 common elements with a point lying on a given line of a system z' of x 3 and 

 a system Z of a 1 right lines with a point on it. If we take for Z 1 the tangents 

 with point of contact of the system of curves (? , vj») and for Z the tangents with 

 point of contact of the system of surfaces (^ , v), then 



p'o = f f p'g' t = j, t G = v , p~g e = a , 



whilst xp* 1 is the order of the surface of contact. 



4 ) Locus of the points, where the surfaces of the three systems have a common 

 tangent. 



5 ) System of v_\ curves of which ^, pass through a given point and v x touch 

 a given right line. 



