356 PROCEEDINGS OF THE AMERICAX ACADEMY. 



all the carves which pass through a given j-.iat x will generate a 

 variety of order eight and dimensions eight 1'^,^, which has the tan- 

 gent space TTi for vertex. 



Then in >S4 



All the CxSivh'ich contain a fixed line p of Y^ and tivo fixed lines in- 

 finitely near to it are contained in a Ci. 



There is a C^ uhich contains all the lines infinitely near, to infini- 

 tesimals of the second order, to a given line of Vi in a given direction 

 (direction being detined by two lines infinitely near of the first order.) 



If the directions are varied, the C^s above will generate a five-para- 

 meter family of lines of order eight which has the lines of a ruled surface 

 Cxfor double lines. 



9. Noteworthy directions on V4. Through each point of F4 pass cc^ 

 important directions. In these directions there are curves whose oscu- 

 lating plane has four consecutive points in common with the Vi. If 

 this is the case, the points (1), (2), (3), (4) must lie in the same plane. 

 If dui and d2Ui are held fixed for the moment and the d^tti are allowed 

 to vary, it is at once seen that if these four points lie in a plane then it 

 is necessary that the points 



(1), (2), (3), "Ifjk duidUjdu^ + 3 ^fT^duidhij, = 0, 



l,fd\ = 0, 



lie in the same S3, and consequently the points 



(7) /=0, f = 0, fr=i\f = 0^ f, = 0, 



(6) ^fikdiiiduj, = 0, 



(8) ^fijjc dUidUjduj, + 3 ^fj^duidhik = 0, 



will lie in an S^. Conversely it is easily shown that if these points lie 

 in an S^ then values of d^Ut can be determined so that the original four 

 points (1), (2), (3), (4) will lie in a plane. Now, writing the last term 

 of (8) in the form 



d^uxlfuduk + d^u^^fkduk + d'Us^f3kdUf,-\- d^Uilfij^dUf^, 



and noting that (6) can be written in a similar form where the d^Ui are 

 replaced by dui we conclude that if the points (6), (7), (8) lie in an S^ 

 it is necessary that the ten points 



(7) /=0, f = 0, f = 0, fs = 0, A = o, 



(9) ^f,du, = 0, /?■=!, 2, 3, 4, 



(10) l.fjkduidujduk = 0, 



