Double Products and Strains in Hyperspace. 43 
the corresponding strain. The interpretation of the form in this 
case gives a quadratic cone issuing from the origin and cutting the 
plane at infinity in the quadratic locus represented by the form in 
the x homogeneous variables. Smith has shown that any trans- 
formation with an invariant quadratic form may be resolved into 
two reflections. From this it is evident that the matrix of any such 
transformation must satisfy a reciprocal Hamilton-Cayley equation 
and that the invariant numbers corresponding to a pair of reciprocal 
roots must be equal. It may be noted that it is not true to say 
that any strain which leaves a quadratic cone issuing from the 
origin invariant is resoluble into two oblique reflections; it is nec- 
essary to add that the strain is unimodular or that the form which 
represents the cone is invariant. 
The question naturally arises whether every projective trans- 
formation which is compounded of two reflections always has an 
invariant quadratric locus, that is, whether the conditions stated for 
resolubility into two reflections are both necessary and sufficient 
for a transformation with a non-degenerate quadratic form in 
homogeneous variables. The answer is negative. To show whether 
any transformation resoluble into two reflection leaves a non-de- 
gerate quadric form invariant, it is merely necessary to examine 
the different cases that may arise. Consider the transformation 
: : 1 ye 
written in the reduced form (76’). Let a@ and © be a pair of roots 
corresponding to no shearing. As far as they are concerned the 
transformation may be written as 


1 4 
ssa MU ee ae UE ag mye 0's NT) =") Mak Bo a 
and the quadratic terms 
[ie ee reli 
on Oe a eo | in OS |e 
ane | 0 | 0 | ee | 0 
0 Ovi eo OE 1 
1 0 0 0 | 0 0 
0 1 0 0 | 
| 0 0 1 0 | 
of non-vanishing determinant are invariant. If there are shearing 
terms, the quadratic terms 
