428 MOORE AND PHILLIPS. 



represent any other point collineation. Then the double product 



' [AB] [a/3] 

 transforms any line L into a line or complex 



L' = [AB](a^-L) = - [BA\{a-fiL). 



Since (a/3L) is a pure regressive product. Furthermore, this is equiva- 

 lent to 



- B-Aa-[BL] = - [B-m = [BL-^\ 



since Aa is the idemfactor. This complex is determined as follows. 

 Let X, Y be two points on L and X' , Y' their transforms by $. Join 

 X to Y'^ and Y to A''. Then by the general theorem of the preceding 

 section L' is a linear function of the two lines thus obtained. That 

 is, these lines are polar lines with respect to L' . This is true whatever 

 pair of points A', Y are taken on L. This proves the following geo- 

 metrical theorem. Let A", 1, Z be any three distinct points on L 

 and X' , Y' Z' , three distinct points on any other line. A dyadic 

 B^ can be found which will transform A", Y, Z into X' , Y' , Z' . There- 

 fore the three pairs of lines XY' , YX'; XZ', ZX'; YZ', ZY' are 

 pairs of polar lines with respect to a complex, namely, the complex 

 into which [AB] [a^\ transforms [A'F]. This is the generalization of 

 the theorem of Pappus for the hexagon inscribed in two lines in a plane. 



The dyadic [AB] [a0\ will represent a collineation if and only if 

 every line XY transforms into a line X' Y' cutting it. The collineation 

 JS/3 then gives a transformation of lines which is a null system. By 

 §14 B^ then determines an involution. 



We can write B^ in the form 



B^ = B[CDE] = B,[CiDxEi] + B^^CoD-^E.] + BslC^DsE^] 



+ B,[CJ)iE4l 

 Then 



[AB][a^] = [AB][a-CDE]. 



= [AB]{{aE)[CD] - {aD)[CE] + {aC){DE]} 

 = - B-Aa-lE[CD] - D[CE] + C[DE]} 

 = - [BE] [CD] + [BD] [CE] - [BC] [DE]. 



This gives the dyadic in a form that does not involve the idemfactor. 

 The result can be obtained from 



B[CDE] 



