430 MOORE AND PHILLIPS. 



If then three of these numbers are zero the fourth will be zero also. 

 This is Pasch's theorem ^^ that if a coUineation represented by B0, 

 with scalar invariant zero, transforms each of three vertices of a tetra- 

 hedron into a point of the opposite plane, it will transform the fourth 

 vertex into a point of its opposite plane. 



Let Aa and 5/3 both be idemf actors and let L = {XY) be any line. 

 Then 



[AB]{a^-L) = [AB](a^-XY) 



= [AB]{(aX)m) - {aY)(fiX)} 

 = A{aX)-B^-Y -\- B(pX)-Aa-Y 

 = \XY] + [XY] = 2L. 

 Hence 



[AB](ap-L) = 2L 

 and so 



[AB] [a^] = 2pq 



where pq is the two-two idemfactor. The double product of the one- 

 three idemfacior with itself is thu^s twice the two-two idemfactor. 



The double product of the idemfactor Aa and a one-one dyadic CD 

 is the line or complex 



[CA] [Da] = -C-Aa-D = - [CD]. 



We have called this the complex of the dyadic. The general theorem 

 states that if CD transforms the planes of the tetrahedron Xi, Xi, 

 Xz, Xi into the points Yi, Y2, Y3, Y4, the complex [CD] is a linear 

 function of the four lines [A\7i], [A'2F2], [Z3F3], [X4F4], that is, the 

 two lines cutting these four lines belong to the complex [CD], 



The double product of the three-one idemfactor aA and a dyadic 

 CD is 



iCa)[DA] = DC-aA = [DC] = - [CD]. 



The interpretation of this coincides with that given in §12. 



If [CD] is zero, the one-one dyadic CD represents a polarity. Thus 

 the condition that a one-one dyadic represent a polarity is that it be 

 apolar with the one-three or three-one idemfactor. 



The double product of CD and the two-two idemfactor pq is the 

 three-three dyadic 



$ = [Cp] [Dq]. 



16 Loc. cit. 



