( 507 ) 



(ƒ/, r). IC lor /■ WO siihslitnto licrc 2(>; — ^1) and for ;<, r the ahovo- 

 iiieiili(iii<'<l iiiiinl>(M>, llir iiimilxM' 4(// 2)(2// — 'OU'^'- 1), appoars a.^aiii. 



3. Let us still cojisider llie correspoiKleiice Wetweeii a point A' 

 and a point l\, l»elon,U'in<>" to tli(> uroup of A', tive points of which 

 lie -with A' on a ("'. Kach point A' is eonjii.uate to (2// — 5)(.v^ — J), . (>•—."); 

 jioints /-*„ ; i-ev(M-sely is — 1), . (2yz — 5) points A' correspond to /V 

 If two conjugate points coincide \vc ha\e evidently a conic l)eai'inii; 

 six points heloniiinu' to a selfsame «ironp of I\ As each of those six 

 points can he regarded as a jtoint A' the nnrnher of those conies is 

 oqnal to the sixth part of the nnniher of coincidences of (lie corres- 

 pondence (/^ A'), thus erpial to (2// — 5)(.y — I);. 



4. If every gronp of an Z" contains less than 5 points, there is 

 no indicated system [("]• In that case we can take (5 — s) arbitrary 

 points Ah, /• = 1 to (5 — xj, and join these by a C^ witli the 

 .V points of a gronp of the I'. To tind the characterizing number [i 

 for the system [("'] obtained in this way, we consider the conies passing 

 through the points .!/, and moreover thi-ough the arbitrary i)oint.l„. 



In 



They intersect C" in the groups of an imolution ls-\ of oi-der 



"i "-J 



2/; and rank (x — 1). Now two involutions //,, and //,-., have according 



to a theorem of Le Paige'), (»j — l\)k.{^i-i — ^<-\)h groups of {l\ -f /•.,) 



2« s 



points in common. Applying this to the involutions Is—\ and U^ we 

 find that through .!„ pass {^2n — s ^1) conies containing each a group 

 of the [\ So n = {2)i~s f 1), v = 2 (2y/— x + l) and d =: 3 (2»— .>-hlV 



5. For .s" = 2 three fixed points A^, .!,, .1, J^i'e wanted. Tiie 

 pairs of liiies of [_C'^^^ form two groups. A ligni-e of the first group 

 consists of a right line Ak Ai and the line connecting A,n with 

 the point which forms with one of the ii points of intersection of 

 C" and A;, A I a pair of the I\ In a figure of the second group the 

 line containing a point Aia bears a })air of the /'. The nnmber of 

 pairs situated on rays through A,n amounts to {ii — i). So we find 

 6='èn-\-'è{n— l)^='^{2n — 1), in correspondence with the general 

 result given above. 



For .s' = 3 we have to take two fixed j)oints A^, .1.,. The pairs of 

 lines form three groups. In the first place there are {n — 3) col- 

 linear triplets (see § 1), of which the bearers through A^ A., are 

 completed to a pair of lines; secondly each point of intersection of 



1) Sur le nombre de? groiipes communs a des involutions supérieures, marquees 

 sur un même support (Bull, de I'Acad. Royale de Belgiquo, 3e série, t. XI, p. 121). 



