MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S^. 353 



there is an unlimited number corresponding to this direction. But an 

 Sq may be determined in this case which hyperosculates four points. 

 There are sixteen such >Sc's. 



In ruled space of four dimensions wc have : 



All the linear .series Ci de/ined by taking a line p of V^ and four lines 

 consecutive to it, if only the last of tkefoii/r consecutive lines is allowed 

 to vary, are contained in a linear series C^. 



The tangent congruence to V^ {the base of the oc* linear systems C5 

 tangent to Vn in p) has three-line contact with all the ruled surfaces 

 passing through p in one of the sixteen principal directions. There is a 

 C3 which has four-line contact with all the surfaces traced on Fg tangent 

 to a principal direction. There are sixteen such series C3. Each one 

 contains the tangent congruence. 



A curve all of whose tangent lines lie in ^ we saw represents a devel- 

 opable surface in /S'4 ; hence, The lines of F5 can be grouped into gc* 

 developable surfaces in sixteen icays ; that is, sixteen of these developables 

 pass through each line.^ Each tangent plane to one of these developables 

 osculates Tj. (The pencils of lines lying in the tangent plane and 

 having their vertices on the edge of regression contain three infinitely 

 near lines of V^.) 



There are sixteen curves of V^ (curves enveloped by lines of J\) having 

 a given line pfor tangent line such that the osculating planes containing 

 p determine the sixteen osculating pencils of which p is a part. 



7. Other important directions. Equations (14) set up a corre- 

 spondence between two directions t and t' defined by dui and hi^. 

 This correspondence is reciprocal, and we saw that the coincidences 

 define the sixteen principal directions. This correspondence between 

 t and t' , however, is not projective, as can best be seen as follows. 

 Consider F5 as the intersection of four hypersurfaces in ;%. If the 

 tangent space ttj is cut by an arbitrary hyperplane t and t' are repre- 

 sented by points in this >S4, the tangent hyperplanes to the hypersur- 

 faces at points of F5 cut V^ in V^'s, having a conical point. The cones 

 formed by the tangent lines at these conical points are represented in 

 the /S'4 (intersection of ttj by a hyperplane) by quadrics. Then t' is 

 the polar of t with respect to these four quadrics ; hence, 



When t describes a line, t' describes a curve generated by the intersec- 

 tion of corresponding hyperplanes in Si of four projective pencils and 

 therefore is a curve of order four. 



^ This system of sixteen developables does not contain all the develop- 

 ables which may pass through a given line, but there are no others having 

 the properties mentioned above. 

 VOL. XLVI. — 23 



