Double Products and Strains in Hyperspace. 23 
The value of this scalar depends only on the dyadic and not at all 
on the particular representation which has been used. This is due 
to the fact that both dyadic and combinatorial products obey the 
distributive law. Such scalars may be obtained regardless of the 
class to which the dyadic belongs. In particular, the double powers 
yield the x scalars 
(33) LANA: Shiicad  eatae aie ga 
where the subscript s has been omitted in the case of ®, which 
is a scalar. 
The scalar of the conjugate of a dyadic is the same as the 
scalar of the dyadic: for the negative sign which sometimes enters 
into the definition of the conjugate occurs in precisely those in- 
stances in which the reversal of the order of the factors would 
introduce a change in the sign of the combinatory product. The 
scalar of the product of two dyadics satisfies the equation 
(34) (PP), = P* B, 
Perhaps the easiest way to see this is to consider both @ and ¥ 
expanded into a block of m? terms of the form (9) where the ante- 
cedents and consequents are reciprocal sets and are the same for 
both expansions. Then (# #); is obviously the sum of the products 
of pairs of coefficients symmetrically situated with respect to the 
main diagonals, one taken from one of the dyadics and the other 
from the other. The same rule applies for evaluating #< ¥%, and 
hence the two expressions are equal. It may be seen directly, or 
by the application of the rules for conjugates and double pro- 
ducts, that 
(35) (PO),=(@P),. 
A more general theorem is that the scalar of the product of any 
number of factors is unchanged by a cyclic permutation of the 
factors. The proof in the case of three factors is contained in the 
equations 
(35)  (PPQ),=[( P) 2), = ¥) 2 Q=QAey (PV) = 
QE (PP), = (2G P),; 
the proof for a greater number of factors is by induction. This 
result may be used to put the matter of invariance of the scalar 
of a dyadic in a different light. Consider any linear transformation 
of coordinates. This may be represented by a dyadic ¥% Under 
this transformation, the strain represented by # becomes 
(36) 2= Pop 
and 25 = (EP BP), =(@ PAP), = By, 
and hence it appears that none of the w scalar invariants of any 
