434 MOORE AND PHILLIPS. 



which shows that [CC] [DD'] is apolar to Aa. It follows that it is 

 also apolar to aA and j^Q- 



We have already noted that a one-one dyadic which represents a 

 polarity is apolar to all the idemf actors. The same is true of a 

 three-three dyadic. 



If rs is apolar to the idemfactors, the double product of rs and -a 

 one-one polarity CD is apolar to the idemfactors. For 



[Cr-Ds] = [{Cr-s)D] - {Cr-D)S = 



Similarly we can show that the double product of rs and a three-three 

 polarity is apolar to the idemfactors. 



If rs is apolar to the two-two idemfactor jyq, its scalar 



{rp)(sq) = {rs) = 0. 



In this case the general theorem states that if rs transforms each of 

 five edges of a tetrahedron into a complex to which the opposite edge 

 belongs, the same will be true of the sixth. 



We ha\e thus shown that if any dyadic is apolar to the one-three or 

 three-one idemfactor, it is apolar to all the idemfactors and, excepting 

 the case of the two-two apolar to the two-two idemfactor, if a dyadic is 

 apolar to any idemfactor it is apolar to all. Furthermore, if two dyadics 

 are apolar to all the idemfactors their double product {if it is a dyadic) 

 is also. 



18. Dyadics symbolically derived from a given dyadic. If 

 we write a one-three dyadic symbolically in the form 



B[CDE] 



we have seen that its double product with the one-three and the two- 

 two idemfactors are 



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

 and 



[BCD]E - [BCE]D + [BDE]C 



respectively. These can be considered as obtained by a sort of 

 symbolic multiplication of B and CDE analogous (except for a pos- 

 sible change of sign) to the outer multiplication. 



Similarly, from any dyadic a series of dyadics are obtained. These 

 are all double products of the original dyadic with the various idem- 

 factors. 



