Double Products and Strains in Hyperspace. 5 
~I 
Q is itself resoluble into two reflections, then it must be possible 
to find a reflection # of type 1 such that the product 2 @ is still 
resoluble into two reflections, and consequently the conditions (102’) 
must be capable of fulfilment in this case. 
The results of the second part of this paper may be summarised 
as follows: 
1°, The determination of the square roots of the idemfactor by 
means of the properties of the equation of least degree. 
20, The determination of the necessary and sufficient conditions 
that a dyadic be resoluble into the product of two square roots of 
the idemfactor. 
3°, The correlation of these results with the theory of reflections 
in connection with unimodular strains and with collineations. And 
in particular, the fact that not all products of two reflections leave 
a non-degenerate quadric invariant. 
4°, The introduction of invariant dyadics 7 and their application 
to the problem of finding the scalar invariants of the product 2@. 
5°. The fact that in case the Hamilton-Cayley equation of 2@ 
is the equation of lowest degree, there may be found a ® = /—20\6 
such that the scalar invariants of the product are arbitrary. The 
corollary that in such cases, if 2,—+41, 2 be written as the 
product of three square roots of the idemfactor with the appro- 
priate interpretation in the theory of reflections. 
6°. The determination of the necessary and sufficient conditions 
that a dyadic be resoluble into the product of three square roots 
of the idemfactor, with an example to show that it is not always 
possible to take a square root of type 1 as the first of the three. 
Massachusetts Institute of Technology, 
Boston, Mass., December, 1907. 
