Double Products and Strains in Hyperspace. 39 
Any term A; ; x; yj; may arise only from the possible combinations of 
Ke =A +H, Hep = aX + KH 
P tl F 1 
and VSS iy Pr i 7. 
If A;; x; yj; is to be an invariant term, the relation 
1 
(78) Aij+ 7 Aity 5+ a Ai, 541 + Atty iH = Ai 
must hold. If this be applied to any of the zero terms, it is seen 
that they yield nothing. If it be applied to the terms in the main 
diagonal it is seen that they an invariant. If it be applied to any 
of the terms in the diagonal next above the main diagonal, there 
is established a set of conditions imposed upon the terms of the 
main diagonal, namely, 
Ai,n:Aan >Asn :Aan 3...=1:—a?: at: —a’:. 
2 me Bare ie 
If it be applied to the terms in the next diagonal line, these arises 
a condition to be imposed on the coefficients in the diagonal next 
to the main diagonal, and so on. These conditions are such that 
they may obviously be solved for the ratios of the coefficients in 
the successive diagonal lines. The result in case of six variables 
X1, X9, %3, Vi, Yo, Y3 gives a quadratic form of the type (stars in- 
dicate the possible presence of terms) 

of which the determinant is clearly not zero; and a similar form 
may be written down for any even number of variables. Now, 
Smith? has shown that the transformation of a quadratic form with 
itself may always be resolved into the product of two involutory 
transformations. Hence the dyadic (76) may be factored into the 
product of two square roots of J. 

1 P. F. Smith, On the linear transformations of a quadratic form into 
itself, Transactions of the American Mathematical Society, volume 6, 
pp. 1-16 (1905). The theorem here referred to is found on p. 13. The 
more detailed exposition of the relations between collineations and strains 
is taken up in our next article 16. 
