MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S4, 359 



In a thre.e-parameter family of lines all the lines infinitely near a 

 given line to infinitesimals of the second order lie in a three-parameter 

 family of order four which has the lines represented by the intersection 

 ofiTz with ^ for double lines. 



In general there are no osculating planes of Fg which h}T)evosculate. 

 Hence in »S'4 four consecutive lines of a three-parameter family are not 

 in general cut by a fifth line. 



13. Special directions. There are oo* directions such that hyper- 

 planes can be found which are tangent to Fg in three consecutive 

 points. Let the three points be 



a^ a:' = X + dx, x" = x + 2dx + d^x. 



If the hyperplane (i, .r) = is tangent at x, 



(') «-)=«. 0-:-^)=«. (^-3=«' 0'3=«- 



and if it passes through x', 



(^, X + dx) = 0, 



which from (1) is seen to be true. Now, if (i, x) is tangent at x\ 



( 



d{x + dx) \ _ 



which on expanding and applying (1) reduces to 



If the hyperplane passes through x", it is at once seen that 



which is seen to be satisfied if (2) is satisfied ; that is, if the hyperplane 

 is tangent at x, it passes through x". If it is also tangent at x", 



(t,^{x+'ldx + d\r)\ = 0. 



Expanding and using the previous relations, this reduces to 



( ^' 2h~t~^ — duiduj ) + 3 ( I, ^. ' - duid\ ) = 0. 

 \ ^aUidUjdUk j \ ^^oUiOUj \J 



