Double Products and Strains in Hyperspace. 57 
+ ay| a's pray Cg ect soc Omi +1| O'ms+2 ae 
ats B1| Bo = B2| B's + oe te Pmi+1|P'ms +2 + se 
where the shearing terms in the last two rows may or may not 
occur; and such a dyadic will evidently satisfy the relations (73), 
It remains to ascertain whether 2 is resoluble as desired. 
The answer is negative. For suppose that 2 = #4, where @ 
and ¥ are involutory. It has been seen in article 12 that the 
equation of lowest degree is the same for the set of dyadics which 
are the transformeds of a given dyadic. Now if @ satisfies the 
equation 
(75) 2° —b, 2P—1 +- by QP? — ... bp B* Ebr Bt1i=0, pn, 
of least degree, so will ©2._@—!, But as @ and ¥ are involutory, 
207 —oOr=e2 
and 
(75°) (2-1)p—b, (Q—1)P-! + by (Q—1)P- 2 — ... 1 Op_-s (Q-')? 
bp 12! Se lia) 
Hence, to extend the use of the term reciprocal to equations in 
dyadics, it may be stated that if a dyadic is the product of two 
square roots of the idemfactor, its equation of lowest degree is re- 
ciprocal. This is stating more than equations (72): for the dyadic 
(74) would not in general have an equation of lowest degree which 
was reciprocal. If the equation of lowest degree is reciprocal, the 
1 : : E 
factors 2—a J, QL which correspond to a pair of reciprocal 
1 : 
roots a, — of the scalar equation must enter to the same degree. 
Moreover, from the results of article 13 it is seen that the invariant 
numbers &, k, ko, ..., Rp—1 are the same for a dyadic and its trans- 
formed dyadics. It is therefore clear that the invariant numbers 
é i : 
which correspond to two roots a, 7 Tust be equal in case the 
dyadic is the product of two square roots of 7. The question now 
is whether these conditions are sufficient for such a resolution. 
onside: the spaces it (a, 4; >=... Gmin fu B2s1--- 4 Ors) ane 
S(C@mi-+1, +++, Bmi4.1-.-), made up of the antecedents which cor- 
respond to any root and its reciprocal and of all other antecedents. 
These spaces are fixed and moreover the space R and the space 
S are independent and together contain m independent directions. 
The transformation in two such spaces will determine the trans- 
formation in all space. But the transformation in each of these 
two spaces is such that its scalar equation would also be reciprocal. 
If now the transformation in these spaces of dimension less than 7 
