( 493 ) 



the plane a^, the node of tliat curve the point O, of' the fii'st generation. 



8. We now ask what arises when the planes «^ and «^ ^i'® 

 placed m such a way in Si\, that the points Qj and (^s coincide and 

 therefore each line drawn through the point of coincidence P^^ in 

 «J is to be regarded as line a^. We then find that to every line a^ 

 through Pj2 in a^ a definite line a^ through P^^ iu (c^ corresponds, 

 so that there is an intinite number of planes «. The locus of these 

 planes « is a quadratic conic space with P^.^ as vertex ; for the 

 pencils of raj^s of the lines a^,a^ throngh P^^ corresponding to each 

 other in «,,«2 are evidentl}^ projectively related. This quadratic conic 

 space must contain, as it contains all generatrices of 0", this tortuous 

 scroll itself. 



Now that the generatrices of this jiarticular surface 0", being the 

 surface indicated in the title, group themselves into triplets lying in 

 a plane, there must be a locus of nodes. This is of order four. If 

 namely we project the surface (P by means of the just found (pia- 

 dratic conic space out of P^^ on to an arbitrary space not containing 

 7^12, the projection is a quadratic scroll having the projections of the 

 planes « as a system of generatrices. Of this surface if the projec- 

 tions of «1 and «2 form thus two lines of the other system ; for each 

 of those two planes has a line in common with each of those planes 

 « and from this ensues that the sections of «, and a, with the space 

 of projection must have a point in common with the sections of the 

 planes « with that space of projection. So in that space of projec- 

 tion each plane through one of the two lines contains a line of the 

 system corresponding to the planes a and therefore the projections 

 of four nodes, namely one on the first liiie and three on the second. 

 So the projection of the nodal curve out of P^^ on to the assumed 

 space of projection is a curve of order four lying on CP, which has 

 one point in common with each of the generatrices of one system, 

 and three points with each of the generatrices of the other system. 

 So the nodal curve itself is a tortuous curve of order four ; it is 

 rational as its projection is. 



Considering the surface 0'' we see at the same time that the sur- 

 face (>" admits of an infinite number of planes cutting it according to 

 a rational cubic curve, namely each plane through P^^ and one of 

 the lines of the system to which the projections of «, and «, belong. 



So we find the following theorem : 



"If we assume in two planes a^ and a^ two projectively related 

 "rational cubic curves, if in these planes we determine the vertices 

 "Qi, Q2 of the corresponding central triple involutions on those 



34* 



