MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S4. 351 



and if this /Sj is tangent to V^, 



(-) G-.^)-'' O'.of)-- «)-■ 



and a similar set of equations for the hyperplanes ^", ^'", ^"^. The So 

 infinitely near is determined by 



(11) (Sr, ^) = 0, (8$", a-) = 0, (Sr", ^0 = 0, {8i^\ ^0 = 0. 



These two spaces will intersect in a line. If the point dx infinitely near 



to .r is to lie in (11), 



(12) (Sr, dx) = . . . (W^, dx) = 0. 



Now, using 8ui to denote the direction of the line of intersection of (10) 

 and (11), and dui to denote the direction of the line joining ,r and 

 a; + dr, equations (12) take the form 



(13) (21^, 2 1 <^«,)=2(& :-!)««*'=«• 



and similar equations for the other ^'s. 



Differentiating (9) and (10) partially with respect to the m's, we have 



/af dx\ f d'\r \ _ 

 \dUi dujj \ ' dUidUjJ 



Substituting these values in (13), 



<») 2(f'.a-£)««.*'^ = (i'.2aS;;;) = o- 



Now, if we put dUi = Suf we have the system of equations (8) ; hence 

 that is the condition that should be verified if the two consecutive tangent 

 spaces are to intersect in the line joining the points of contact. 



Since ]\ is contained in <£> (the intersection of the five quadrics fl^), 

 the tangents to l\ which have three-point contact must lie entirely in 

 ^. The sixteen lines discussed above then represent pencils of lines in ^S'^ 

 and may be called oxculdt'mg pencils, since they contain three lines of F5 

 infinitely near. Further the planes of these pencils are inflexional for 



