200 Mr TurnhuU, Some Geometrical Interpretations 



Again, ^ — would give six quantities which would symbolise 



the coordinates of a certain linear complex : and, m some special 

 cases, the coordinates of a straight line. For example, 



is a useful way of denoting the six quantities (Aj)) A^ {i,j = 1,2, 3, 4), 

 which represent a straight line, since they satisfy the identical re- 

 lation existing between the six ^^-coordinates of a straight line. 



Li7ie coordinates. 

 § 9. This identity satisfied by line coordinates (p) is 



^pijPM=0 (1), 



which we denote by co (p) = 0. Symbolically, the condition that 

 two lines p and q should intersect is {pq) = 0. If p is the line 

 common to two planes u, v, and q is that common to u', v', then 

 this condition is (iivuv) = 0. 



If two lines p, q intersect, then Kpij + Xqij represents the co- 

 ordinates of any line of the plane p, q passing through the connnon 

 point of p, q. Since the line p touches the quadric / if {Ap)- = 0, 

 it follows that the line (k, \) touches this quadric if 



k' (Apf + 2k\ (Ap) (Aq) + X2 (^Aqf = 0. 



Hence (Ap)(Aq) vanishes if p intersects the conjugate of q in f; 

 for then p and q are harmonic conjugates of the two tangents to / 



r)TT 



in this pencil of lines (k, \). This shews that the coordinates ^::— , 



op 



i.e. (Ap)Aij, are those of the line conjugate to p in the quadric f. 



Analytically it is evident that these coordinates represent a line 



and not a linear complex, since they satisfy the required condition 



(1). In fact 



(Ap) (A'p) (A A') = A {AAy <o (p^. 



But the left member of this equation is the symbolic equivalent of 

 substituting (ApJAy forp in (1): which proves the statement. 



Complexes and their polars. 



I 10. Let (Dpy = represent one of the quadratic complexes 

 of § 7. Then (Dp) By gives the coordinates of a linear complex 



* Cf. Gordan, ii, § 6. 



