MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S^. 357 



will lie in an Sg, and conversely, if these ten points lie in an S^, d-iii 

 and d^Ui can be so determined that the original four points will lie in 

 a plane. The condition that these ten points lie in a plane is 



(11) \x ~ . . . ^ VikC^u^ • • • '^/ijjihdujdui, I = 0. 



OUi Oil^ 



Hence, if the dui be considered as the homogeneous coordinates of the 

 directions of the tangent lines at a point of 1^4, we see that Through 

 each point of \\ pass cc''* directions whose tangent lines form a cone of 

 order seven such that along these directions there are curves whose oscu- 

 lating planes cut V^ in four consecutive points. 



Then in S^ we have 



Through each line p of a four -parameter family F4 pass oo^ directions 

 such that there is a ruled surface tangent to each of these directions (con- 

 taining the two infinitely near lines which determine the direction) and 

 possessing the property that four infinitely near generators are cut by 

 one line. 



It is known that four lines determine a Cq (the intersection of six 

 Cs's), but through four infinitely near lines of one of these ruled surfaces 

 pass a whole pencil of Co's. There are Ci's which have five lines infi- 

 nitely near in common with F4. 



A four-parameter family of lines can he generated by cc^ ruled sur- 

 faces such that four consecutive generators are cut by a line. This can 

 be done in co^ dij^erent ways. 



10. Another set of important directions are those along which a 

 tangent C4 to F4 has double contact. In S^ a tangent /S7 to a F4 is 

 defined by the equations 



(12) 





If this >S^7 passes through .r -f da:, 



(^, a- + d.r) == 0, (f , a- -f d.v) = 0. 



Expressing d.r in terms of dui and applying (12), it is at once evident 

 that these equations are satisfied. Now, if S^ is also tangent at this 

 point, 



