350 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(tt, /5, y, 8 are the coordinates of four hyperplanes passing through ttj). 

 The cones therefore will form a linear series of "three parameters and 

 hence will have sixteen generators in common. These are the sixteen 

 directions upon which rest many of the infinitesimal properties of higher 

 order of five-parameter families of lines. 



The tangent space ttj cits T's in a curve which has a multiple point 

 of order sixteen in x. The tangents at the multiple point are the 

 sixteen directions noted above. 



These sixteen directions correspond to the asymptotic directions on 

 a surface in S^ both in the sense that two successive tangent spaces tts 

 intersect in a line joining the points of contact and in the sense that 

 tangents to V^ in one of these directions have three-point contact. The 

 latter property can be seen by deriving the condition that a tangent 

 line have three-point contact. From equations (4), (5), (6) we see that 

 this would simply lead to the condition that the point 



S fikdutduk = 



should lie in the tangent tts, because in that case the quantities d^Ut 

 could be so chosen that the three-point (4,) (5,) (6), that is, (x), (dx,) 

 (d^x), would lie on a line. The condition is, then, the vanishing of the 

 determinants of the matrix 



dx dx dx dx dx -^ dx , , 



* 5— ^r^ ^~ H~ £^ ^ ^—^—duidUk 

 oui d«2 dUa dUi dWj "^^ dUidU/^ 



and this is exactly the condition that defines the sixteen directions 

 above. 



5. In order to see that the tangent spases at two consecutive points 

 of V^ along one of those sixteen directions intersect along the line 

 joining the points of contact, we will first examine what corresponds to 

 conjugate directions on ¥5.^ If we consider the points of V^ as deter- 

 mined by the hyperplanes passing through them, then any point will 

 be given by 



a, x) = 0, 



where a:* is a function of the five «'s. Then, if the tangent -r^ be looked 

 upon as the intersection of four hyperplanes, the condition that the 

 hyperplanes pass through x is 



(9) a>') = ^. (r,.r) = 0, {r,x) = 0,(e\x) = 0, 



* The arRumcnt hero is the same as that used by Pogre for discussing a 

 Vs in Sn in his 1007-8 lectures on "Geometria deUa rctta." 



