MOORE. — INFINITESIMAL PROPERTIES OF LINES IN S4. 347 



The intersection is of order three and dimensions four. The lines 

 which cut two fixed lines Avill be represented by the intersection of two 

 such tangent spaces ^^ with <1> ; this is an ordinary quadric contained 

 in <I>. 



2. Directions in ruled space of four dimensions. In ruled space 

 of four dimensions as in three dimensions a direction through a given 

 line of a system is defined as the Chasles correlation which a ruled 

 surface belonging to the system and passing through the given line 

 determines; that is to say, a direction through a given line is deter- 

 mined by the given line and a line infinitely near to it. In the S^ 

 defined by the coordinates joj^ this corresponds to the tangent lines to 

 the curves traced on the variety <J>, but since the points of ^Vg which lie 

 outside of ^ have no significance, the tangent lines in general will have 

 none. We can, however, consider the polar of such a line. This polar 

 will intersect O in an ordinary quadric cone, and since it can be looked 

 upon as the intersection of two tangent spaces aS'c to ^ at infinitely near 

 points of the given line, the points of this cone will represent lines of 8^ 

 which cut two infinitely near lines, that is, will represent the special 

 congruence determined by the two infinitely near lines. This congru- 

 ence represents the Chasles correlation determined by the two lines. If 

 the tangent line to <^ lies entirely in <f», the quadratic cone and conse- 

 quently the congruence will degrade. A curve all of whose tangents 

 lie in ^ will represent a developable in *S4 because all the Chasles 

 correlations are degenerate. 



We saw that a tangent /% cuts 4> in a V^. Then, if this >% is cut 

 by an arbitrary hyperplane, the F/ will be cut in a variety ^3^ of order 

 three and dimensions three. Each point of 2, the intersection of /S'e 

 and the hyperplanes, will represent a direction through the line r corre- 

 sponding to the point of tangency. The points of <^^ will correspond 

 to the special directions through r, that is, to the degenerate projectivi- 

 ties. The variety ^3^ and the space 2 in which it stands can be used 

 instead of the ones used by the author in the paper previously referred 

 to, and all the results there obtained can be obtained here. Properties 

 of lines involving differentials of the first order are disposed of in this 

 manner. 



I. Properties op Five-Parameter Families of Lines. 



3. In the following, for the convenience of v/riting, the coordinates 

 ^1, .2-2 ... ^10 will be used instead oipxi, pu • ■ . Pu- Also a variety 

 on <I> and the system of lines which it represents will be denoted by 

 the same symbols where there can be no ambiguity. The symbols 

 C'l, C2 . . . Cs will be used to denote linear systems of lines. 



