THE DYADICS IN THREE DIMENSIONS. 425 



If / i'.s' the two-two idemjador this is equivalent to 



pi[<J>,1> - \I]p2 = 0. 

 Hence 



Conversely, if this condition is satisfied the dyadic determines a collinea- 

 tion or correlation. 



If a collineation or correlation is set up by either a self-conjugate or an 

 anii-self-conjugate dyadic, the transformation is an involution. For 

 then 



$ = ± <S>c 



$.$ = \I. 

 Hence 



$2 = ± x/ 



which shows that two apphcations of the transformation $ gives 

 identity. Hence $ determines an involution. 



II. Double Products. 



15. The double product ^^ of two dyads AB and CB is defined as 



[AC] [BB]. 



In general this is a new dyad. If one of the factors [AC], \BB] is a 

 scalar, it is however an extensive quantity. If both factors are scalars, 

 the double product is a scalar. 



The double product of two d^'adics 



AB = ZA,Bi 

 CB = -ECiBi 

 is the sum of terms 



AB -.CB = 2UiC,] [BiB,] = [AC] [BB] 



obtained by multiplying the two dyadics distributively. Since the 

 products [AiCk] and [BiBk] are distributive, if the antecedents or 

 the consequents of either dyadic are replaced by their values as linear 



15 Gibbs-Wilson, \'ector Analysis, page 306. Wilson's paper quoted above. 



