MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S^. 355 



Through each point of V^ fass oc^ curves such that along these curves 

 hyperplanes have two-point contact with V^. 



Tangent Trg's at two consecutive points of one of these curves do not 

 intersect in a line but in general in a plane. Then in Si we have the 

 following theorem: 



Through any line p of V^ may be passed <x>^ ruled surfaces in such 

 a manner that there are linear series C^ tangent to V^ in two lines of 

 the ruled surface infinitely near. 



II. Four-Parameter Families F4. 



8. Let Vi be the variety on ^ whose points represent the family of 

 lines. The tangent lines to T^4 at a point x generate a linear space TTi 

 of four dimensions. Through tt^ pass 00^ /SVs and hence, There are 

 oc^ linear series 64 tangent to a four-parameter family of lines in a 

 line of the fam ily. 



The osculating spaces of the curves on F4 which pass through a 

 given point x are determined by the points 



(1) /=0, 



(2) l.f,du, = 0, 



(3) ^fikduiduj, + '2fd'Ui = 0, 



(4) ^fjj,duidujdu^ + 3 LfikdUid^Uk + ^fd^Ui = 0, 



(5) l^fijkiduidujdukdui + 6 'Lfjj,dUidUjd^u„ + 3 ^fijd^Uid\ + 



4 S/y dUid^Uj + ^fi dSq = 0. 



By the same reasoning previously used it is at once seen that the oscu- 

 lating /SVs which have the same osculating Ss in common generate an 

 ^7 and the osculating planes having a tangent line in common generate 

 an /S5. 



The iS's generated by the osculating planes having the tangent line 

 (having direction dui) in common is determined by the points 



/=0, f = 0, A = 0, f = 0, f = 0, ^f^du4u^ = 0. 

 It is then the space which joins 774 to the point 



(6) Ifitduiduic — 0. 



Now, if dui are allowed to vary, (6) will generate a variety of three 

 dimensions and order eight ^ V^, and hence the osculating planes of 



® See Segre. "Prelimnai di una teoria della varita luoghi di spazi," Rendi- 

 conto di Palermo, 30, 104 (1910). 



