( 506 ) 



Tlie mimbers of pairs oï lines caji 1m> (Icleriiiiiied in the tbllowing 

 wav. Let P, P' , P" l)e tln-ee poijils ol' tlie same uroiij» of tlie /* : 

 we make P to correspond to eacii of llie points N which the right 

 line P'P" determines on C'- ; as to P Itelong \{s — 'J)(.v- — 2) pairs 

 P',P" each point P in the correspondence [P, S) is conjngate to 

 (.9— l)^.{ii — 2) points S. The pencil of rays having S as vertex 

 determines on C" an /"—i liaving {n — 2)(.y — J) i)airs l*',P" in com- 

 mon with h : so to each point Ncorresjiond {n — 2)(.y— 1)(a' — 2) points P. 

 When now two conjugate points /^. <S' coincide, three points y^ /^', /*" 

 lie in the same right line and each of those j)oints is to be regarded 

 as a coincidence of the corres])ondence {P,S). So the nundjer of these 

 collinear triplets is {ii — 2)(.v — I)». The hearer of sucii a triplet forms 

 with the connecting line of two points belonging to the same gronp 

 a pair of lines of [C']; conseqnentlv 



Ö == (,,_2)(.-l),(.-3), = (3 (//- :!)(. -1),. 



From this ensues again 



f/ = 2 (»— 2)(.s— 1), and r = 4 (;/— 2)(.y— 1),. 



2. On each conic of tiie system [C^] li\e points /^ are lying and 

 {2n — 5) points X more. Eacli point of C'^ can be regarded as a 

 •point P and as a point A. Of the (i conies through that point tiiere 

 are (.s- — !)_,, connecting J* with four points J' l»elonging to the same 

 group; the remaining (2/^ — b](s — 1), contain l)esides A' a quintuple 

 of points of the P and (2^^ — 6) points A' more, which we shall adjoin 

 to A'. The j)oints A', A' evidently form a symmetric correspondence 

 with tiie characterizing numl)er (2/^ — 6)(2« — 5)(.s- — 1)^. Each coincideiice 

 of (A, A') furnishes a conic of [C"], touchijig C". 



Besides these 2(2« — 6)(2??, — 5)(.y— 1)^ conies there is a group of 

 touching conies each of which connects a coincidence of /" with 

 three points behmging to the same gronp of /"; tlieir number amounts 

 to 2(.v-1)(.v-2)3^8(.s— J),. 



Bui there is still a third grou[) of tangential conies. When a point 

 P coincides with one of the points A', the conic touching C" in 

 PzEA' represents two cur^'es; so in that point O' touches likewise 

 the envelope of [6'^]. Xow to each poiiit /-* belong {2n — 5)(.v — 1)^ 

 points A', Avhilst eacli point A' is conjugate to 5(2/^ — 5)(.v — 1)^ points 

 P. Therefore tlie third group contains ()['2n — 5)(.y — 1)^ conies. By 

 counting these donble we arrive at 



\2{2u-i5){2n-b) + 8 + 12 (2«-5)](..-l), or ^n-2){2n-lp-l)^ 

 conies touching C". This number can be easily controlled ; for, a 

 curve C' of class /.', is touched by (/■ jx + /m-) curves of a system 



