528 



in common with tliis curve. The number of apparent double points 



of y is therefore eqnai to 7nm'(2mm' — m — m'). 



If we choose L in the point of intersection B^ of p, and w,, 



mm'iinm' — 1) „ , , , /. , , • ■ ■ , ■ , 

 oi the ciiords ot y through this point coincide with 



each of the lines />, and v.^. Tiirough B^ tiiere pass accordingly 

 mm'(m — l)(m' — J) bisecants of y different from /); and w,. According 

 to § 3 these are the representation of as many plane jiencils through 

 a containing two straight lines of i2, hence also of r. The rank 

 of the congruence r thai troo complexes of the orders m and m' 

 have in common, is therefore equal to mm' {m — 1) (m' — J). 



If we substitute this number for /• in the expression fonnd in § 5, 

 and if we make a and li equal to mm', we find indeed that the 

 order of the surface formed Iw (he vertices of the plane pencils 

 containing three straight lines of the intersection of (wo complexes 

 of rays of the orders m and m' , is equal to: 



I mm' (7?im'— 2) {2inm' —SmSm' + 4). 



We get another check tiirough the application of our formula to 

 the congruence consisting of the stz'aiglit lines passing through one 

 of n given points. For this congruence a = n and ii = r = 0. The 

 locus of the vertices of the plane pencils which three straight lines 

 have in common with this congruence, consists of the planes that 

 may be passed through each triple of the given points. By the said 

 substitutions in the formula of § 5, we tind indeed the number of 

 these planes, namely : 



i n (n— 1) {71-2). 



To the theorem derived in § 5 there corresponds dually -. 



The planes of the plane pencils that have three straight lines in 

 common with a congruence \", ^\ of the rank r, envelop a surface 

 of the class : 



^(/3-2)|6/-— ((i— l)(3«-ii)j. 



I 



