THE DYADICS IN THREE DIMENSIONS. 433 



The double product of B[CD] and the two-two iderafactor pq is 



[Bp][CDq] = [BCD]. 



17. Dyadics apolar to all the idemf actors. If the double 

 product of rs and Aa is zero, we may consider rs as analogous to a 

 polarity. In that case, if 



ny = [CD][EF] 

 [rA][as] = [CDF\E - [CDE]F = 



and hence 



. [CDF]E = [DCE]F, (A) 



Let A'', Y be any two points. Then by direct expansion we get 



{XYEF)[CD] = {XYCD)[EF] + {CDEF)[XY] 

 - iXCDE)[YF] - {YCDF)[XE) 

 + iXCDF)[YE] + {YCDE)[XF]. 



But from (A) we have 



(CDFE) = (CDEF) = 0, 

 Also 



(XCDF)[YE\ = {XCDE)[YF], 

 {YCDF)[XE] = (YCDE)[XF]. 



Hence 



{XYEF)[CD] = (XYCD)[EF]. 



Since this is true for all value of X and Y 



[EF][CD] = [CD][EF]. 



The dyadic is therefore self conjugate and its scalar vanishes. Con- 

 versely, if these conditions are satisfied it is easy to show that rs is 

 apolar to the one-three, the three-one and the two-two idemfactors. 

 It is thus apolar to all the idemfactors. 



The double product of two polarities is apolar to all the idemfactors. 

 For let CD and CD' be two polarities. Then 



[CD] = [CD'] = 0. 



Hence, if Aa is the one-three idemfactor 



[CC-A][DD'-a] = [CC'A]{{Da)D' - {D'a)D\ 



= - [CC'D]D' + [CCD']D = 0, 



