384 HITCHCOCK. 



A"*I = A' r (96) 



or in words: the dot reciprocal is the star product of star reciprocal 

 and dot idemfactor. Similarly by operating on the star idemfactor, 



A",.= A' r :J" (97) 



or: the star reciprocal is the dot product of dot reciprocal and star 

 idemfactor. Thus the star reciprocal of ii is ii: I" ov\ (jj + kk— ii), 

 and the star reciprocal of ij is — ji etc. 



Star multiplication may always be replaced by dot multiplication 

 by introducing the double dyadic /*/ between the factors. For if M 

 and N are any two dyadics we have M: / = M and /: N = N, hence 



M*N = M : 1*1 : N (98) 



We also observe that 1*1 is the reciprocal (double dyadic) of I" with 

 respect to double dot multiplication, that is 



I":I*I= I"*I= I (99) 



In a similar manner double dot multiplication can be changed to star 

 by the operation /": I" for 



M:N = M*7" :/"*N (100) 



and I": I" is the reciprocal of / with respect to star multiplication, 



/*/":/"= I\I"=I" (101) 



It is thus plain that, in spite of the imaginary character of star 

 orthogonality, the reciprocals of a set of real dyadics are real, and the 

 reciprocal, (or inverse), of a real double dyadic, if it exist, is also real. 



15. Illustration of Various Types of Multiplication. 



We may close this discussion of the general properties of double 

 dyadics by a few examples of the types of product considered. Con- 

 sider the two tetrads ijji and jiij. If they be taken as polyadics of 

 order 4 and multiplied by multiple dot product we have 



(ijji) : (jiij) = (102) 



where colon means multiple dot product. 



If they be taken as a pair of double dyadics the scalar (36) is 



((ij | ji, ji I ij))s = ij : jiji : ij - ij : ijji : ji = - 1 (103) 



where the colon stands for double dot product, for we assume the case 

 A* = 2, hence might also have written (ij | ji)*(ji | ij) for this scalar. 



