MOORE, — INFINITESIMAL PROPERTIES OF LINES IN S^. 349 



(7) /= 0, j\ = 0, ./; = 0, A = 0, A = 0, y; - o, 



'^/ik'fttiduk = 0. 



Since there are oc* planes obtained allowing d^iii to vary, this So must 

 be the locus. From (7) it is evident that the Sq contains the tangent 

 space TT., which is determined by 



/=0, J\ = . . ./, = 0. 



AYe will call the space defined by (7) an hypej-osculating /%. Now, if 

 an >% containing ttj is given, then a direction (values of dui) can be 

 determined so that the S^ will hyperosculate V^ along this direction. 

 It is only necessary for the dui to satisfy the relations 



(^, VikdUidUk) = 0, 

 (8) ($', ^fiiMdu^) = 0, 



(^", ^fikdUidUk) = 0, 



where the symbol (^, .r) = is used for E i^^\r^^^ = and the Sq is defined 

 as the intersection of the three hyperplanes 



(^, ^) = 0, (e,x)=0, (r, ^) = 0. 



Then the tangents to Fj along which the S^ is hyperosculating are 

 determined by the values oidui which satisfy (8). There are oo^ values 

 of dui which satisfy (8), and the tangents so defined generate a cone of 

 order eight. Therefore an Se passing through ir^ will hyperosculate V^ 

 along <x>^ directions forming a cone of order eight. This shows that 

 the hyperosculating S^ cuts V5 in- a surface which has a conical point 

 of order eight. It is also evident that a hyperplane passing through a 

 h3rperosculating ^S'e will cut V^ in a F4 which has a conical point of 

 order two, since only the first of equations (8) would have to be satisfied. 

 Similarly, an S^ which contains the hyperosculating S^ will cut F5 in a 

 V3 which has a conical point of order four, since only two of equations 



(8) would have to be satisfied. 



4. We have just seen that a hyperplane passing through ttj cuts V^ 

 in a F4 with a conical point in .r, the point of tangency. Let us now 

 examine how this cone varies as the hyperplane varies. The coordinates 

 of the hyperplane will be 



