1190 



as the remaining two points ') of the qnadrnple lie on ?«, the j^airs 

 W,Q"' whicli complete the pairs Q,Q" into groups of {Q^), will 

 lie on a conic (ij/. As s^. with the enrve ^7//, apart from Sk, has 

 two points in common, Ö// passes thi-ongh the fonr points .S'/ . In 

 the transformation {Q,Q') ■'^k corresponds to the tignre of order 8, 

 which is composed of s/^. itself, (J/-^ and o/,'^ counted twice ; this 

 figure, as it ouglit to do, |)asses three times through the })oints *S'. 



Every singular line s/^ is bitangent of the curve of involution (7)10, 

 mentioned above, for it bears two pairs Q' ,Q" , for which the point 

 Q is intersection of (ik^ witii /. The singular line ?ms septuple tangent 

 of (^i)io' for lirst / cuts the conic t" in two points, which each 

 determine a triplet of the /•' lying on a, on account of which ?ó is 

 six times characterized as tangent; but u contains moreover the pair 

 of points Q' ,Q" indicated by the intersection Q of it with /. 



The curves {q)^^ and {<j)^* belonging to / and /-•■ have therefore 

 in coixunon the line u, which rej)resents 49 common tangents and 

 the 10 lines .v, which each r3present four of those tangents; the 

 remaining 11 we find in the 3 lines indicated by the point //* and 

 the 8 which are determined by the intersections of /* with /** i^cf. § 5). 



The curves a^^ and (>./' have the points 8^,8^,^^, in common and 

 intersect twice in S^ and aS'^, the remaining two intersections V^^ 

 and Fj./ form with ,S'i and .S', a quadruple. From this it ensues 

 that through each two points of q' passes only one curve of [o'']. 



The triangles of involution Q'Q"Q"' described in a^^ envelop a 

 curve of class four (for S^ belongs to two of those triangles); this 

 curve of involution has )i as threefold tangent, for u, bear.s a triplet 

 of points forming with ,S\ a group of the {Q^). So u represents nine 

 common tangents of the curves of involution belonging to *S, and S^ ; 

 the line V^^V^^ is also a common tangent; the ♦•ernaining six are 

 apparently singular lines ,s' and form three pairs, which respectively 

 pass through S^,S^,S^. 



The singular line .y^* is cut by the conic a-]^ in two points, whicii 

 form a quadruple with two points of S]^; so they lie on 0*7:. Conse- 

 quently sj. and .y//" are opposite sides of c>?2é^ quadrangle of involution, 

 which has ;S, as adjacent vertex. The two coincidences of the (Q*) 

 lying in S^ also determine quadruples, for which -Sj is adjacent 

 vertex. It is easy to see that there are no other quadruples of which 

 two opposite sides intersect in ;S\. From this it is evident that an 

 arbitrary point is adjacent vertex of three quadrangles. 



1). One of those points lies on si* and forms with Sk a pair of the P lying 

 on that line. 



