412 DK L. CREMONA ON AN EXAMPLE OF THE 



construct a threefold linear system of sextics K = (123456) 2 , whose tangents 

 at each of the six common nodes are coupled in involution, and which cut, also 

 in involution, each of the six conies [1], [2], . . and of the fifteen right lines 

 12, 13, . . . Any sextic of this system is the Jacobian of a re\seau of cubics 

 123456. 



Among these oo 3 cubics, there are oo 1 curves possessing a cusp (stationary 

 point), and the locus of the cusps is a curve = (123456) 4 of the twelfth order, 

 which touches each conic [1], [2], . . . and each line 12, 13, ... in two distinct 

 points, and has (only) two distinct tangents at each quadruple point 1, 2, ... : 

 those points and these tangents being the double elements of the twenty- seven 

 involutions mentioned above. 



Let us start now from the foregoing plane diagram, without any further 

 reference to its origin ; and consider tt as representative of a surface 4> whose 

 plane sections shall have the sextics K as their images.* We see at once that 

 the order of <[> is 12, for, two sextics K meet in (6.6 — 6.4 = ) 12 more points. 

 Thus we get a (1, 1) correspondence between the points of tt and those of 0; 

 any point M on tt being common to oo 2 sextics K, it is the image of a point M' 

 on <t>, in which the oc 2 corresponding planes meet. But if M lies on one of the 

 six conies [1], [2], . . or of the fifteen lines 12, 13, . . or infinitely near to one of 

 the six points 1, 2, ... , then all the oo 2 sextics K passing through M contain 

 also another common point M x , which is conjugate to M in one of the twenty- 

 seven involutions. Therefore, in such case, M' is a double point on <I> : this 

 surface has an infinite range of double points, whose locus, as easy to see, is 

 constituted by twenty-seven right lines, having as their images on tt the six 

 fundamental points and the six conies and fifteen lines connecting them. 



If M falls at the intersection of 12, 34 viz., if it belongs to two involutions, 

 it will have two conjugate points M 1 ^(12) (56), M 2 = (34) (56) ; and the 

 three points M M 1 M 2 will be common to oo 2 sextics K corresponding to oo 2 

 planes, whose point of intersection M' (where the nodal lines of <!> meet, which 

 answer to 12, 34, 56) is consequently a treble point on <1>. Thus, our surface 

 possesses forty-five treble points, in each of which three nodal lines meet. 



Let a cubic 123456 have a cusp M ; then, every sextic (123456) 2 , which is 

 the Jacobian of a rdseau including that cubic, shall pass through M and touch 

 there the tangent at the cusp. Hence the oo 2 sextics K through M will have 

 the same tangent at this point. Accordingly the corresponding point M' will 

 be a double point on <I> with coinciding tangent planes, viz., a cuspidal or 

 stationary point. Thus we see that <1> has a cuspidal curve, whose image on n 

 is the locus of cusps of cubics 123456, viz., the curve G = (123456) 4 of the 

 twelfth order. The order of the cuspidal curve on <I> is (6.12 — 6.2.4 = ) 24. 



* See CapOBaLI's paper in Collectanea Math, in memoriam 1). Chelini. 



