298 



It possesses sti-en singular points and iive. singular straight lines. 



9. In ^ 7 there was a reference Jo a triple involution tliat has 

 only ('ollinear grou|)s. Another /, with only coilinear triplets is 

 determined by the projective nets 



■ ka,' -f //>,' + mc/ = 0, /;.'], -f- //>', + mCx = 0. 



Each triplet consists of' base-points of a poicil {c") belonging to 

 the net [c*] indicated by 



\ a /J y I 



a/ b/ C:,' =0, 



-"jf J^x ^.r 



whicii has thirteen fixed base-points ;S/^.. For the curves r/^' /i,.^^^.'/1j. 

 and a/Cx = Cx''A.r have in coniuion the three points indicated 

 by a/ =zO, Ax = 0, and they do not lie on the net-curve b/ 6V = c;./ B,c- 

 The curves of [c*] pass therefore through 13 fixed points. 



Any straight line contains tliiee base-points of a pencil (c^). \t' it 

 is re[)resente(l by kAx -\- IB^ -\- nid = 0, which is always possible, 

 the pencil in r|uestion is found by writing 



ka 4- ti^ H- 7ny = 

 in 



k(( -)- li'i -\- my 



kAx-^llh -\-mC, 

 Then we find the pencil 



? 



3 



:s hAx 



3 





7 



and it has as base-points the intersections of 



^ küx' = with :^ kAx = 0. M 



The thirteen points Sl- ai-e singular, for each point S foims a 

 triplet with each of the pairs that is produced by the intei'section 

 of the pencil with centre >S^- on the nodal curve (ik\ which has 

 Sk as node and belongs to [c^j. 



The groups of the /g that are coilinear with the point P lie on 

 a curve {J^)\ which passes through the points *S', consequently also 

 belongs to [c*]. 



1) An arbitrary net [c^| has 12 base-points at most and intersects a straight 

 line in the groups of an involution J^^ (of the second rank), which has three 

 neutral pairs. Here the three pairs are replaced by a neutral triplet. 



