297 



curve Y*, whicli contains the coincidences of the cubic involutions 

 lying on the rays. 



Analogous results are arrived at by considering the net of w^hich 

 the six base-points lie on a conic. 



8. Let a net of nodal cid)/cs be given, which all pass through 

 the base-points B^, B^ and have their node in D. 



To /;, =£^iZ) belongs a [)encil of conies passing through D, B, 

 and two other points A^ and A^^. Analogously to b^zii BJJ a (c*) 

 with base D, B^, A^, A^*. The two pencils [c'] indicated by this have 

 the threeside in common, which consists of b^, h^ and a third line (/. 

 From this it ensues that (^/ must contain the points A^, A^*, A, and A,*. 



On the singular sli'aight line d, [c^] determines an /, ; here too 

 we have consequently a triple involution, which was excluded in 

 the investigation mentioned above, because it has coUinear tri|)lets. 



On the pair of lines DA^, B.^ A*[_c^] determines groups of the /j, 

 which have each a point on DA^ and a pair of points on B^ A^*. 

 The last mentioned line is therefore sinqular, and the same holds 

 good for the lines B, A,, 7i, A, and 7^^ /I,*. 



Taking into consideration that the cnrve of coi?icidences \s a y* wliU 

 triple point I), we can now deduce from the combination of two 

 curves {p)^ that besides the jive singular straight li/ies mentioned 

 there can be no others. For, {p)^ has (/ as bitamjent, so that d 

 represents four common tangents of {p)^ and {q)^. And, as to p, on 

 account of yS a curve // is associated, as locus of .Y' , {p)^ and (^"i^ 

 can only be touched yet by four singulai' straight lines. 



As none of the singular lines passes through D, the curve {Py 

 for F'^ D will have a triple point. On this <i^ which passes through 

 i^,, B^ and (he points A, lies an 7,, of points A. A', for which A'' 

 is lying in D ; the straight line A A' envelops a curve of the 3''' class. 



For B^ the curve {Py consists of a conic (i^^ (which contains an 

 7,) and the straight lines B^ A^, B^ A^*. 



The singular points A^, A* form triplets with each of the points 

 of b^ ; to them no singular straight line is therefoi'e associated. For 

 A^ the curve {Py consists of the straight lines A^ B, and />, together 

 with the twice to be counted line d. 



The curve {p)^ is evidently of order 8 (two bitangents) ; it is conse- 

 quently intersected by p in 4 points. Consequenüy the complementary 

 curve is of the fourth order. As it has nodes in D, B^, B,, it can 

 have besides these points but 16—2 X 3 — 2 X 2 or 6 points in 

 common with '/\ In this /j only three groups occur of which the 

 three points have coincided. 



