66] REPRESENTATION BY LINE -COORDINATES. 223 



Suppose that the two members of a vector -cross have Pliicker's 

 coordinates 



X^Z^M^ and X 2 Y 2 Z 2 L 2 M 2 N 2 



with the identical relations, 



Their resultant has components 



and the volume of the tetrahedron is one sixth of 



LX + MY + NZ = 



which in virtue of the two identities is 



45) L,X 2 + M,Y 2 + N,Z 2 4 L 2 X, + M 2 Y, + N&. 



If any two lines are given by their Pliicker's coordinates, the 

 condition that they shall intersect is that the above expression shall 

 vanish. 



We may now find the equation of the complex of double -lines. 

 We have seen that every line meeting two conjugate lines is a 

 double -line. Let the coordinates of the two conjugate lines be 

 X 1 . . . N lf X 2 . . . N 2 , satisfying the conditions 



46) 



^ Z,, 



where Xf^Y^Z^L^M^N^ define the vector -system. Let the coordinates 

 of a double -line be XYZLMN. The condition that it meets the 

 line X^Z^M^ is 



L t X + MJT+ N,Z -f X,L + Y, M + &N= 0, 

 and that it meets X 2 Y 2 Z 2 L 2 M 2 N 2) 



L 2 X+M 2 Y+ N 2 Z+ X 2 L + Y 2 M + Z 2 N = 0. 

 Adding these equations, and using the conditions 46) we obtain, 

 47) L X+M Y+N l> Z+X i> L + 



as the equation of the complex, that is, any linear relation in 



Pliicker's coordinates represents a linear complex, as stated in 63. 



It is to be noticed that the equation 47) does not signify that 



the line XYZLMN cuts the line X o r o Z L J!f JV" unless the latter 



