Double Products and Strains in Hyperspace. 53 
ing to these roots. The shearing terms will be £—1 in number 
and will be situated in the next diagonal line. In constructing 2(?) 
there will be shearing terms in the corresponding line. Moreover 
the next diagonal line will contain terms to the full number k—2: 
but there will be no other terms in the half square. In construction 
=(3) there will be terms in these two lines and also in the line third 
removed from the main diagonal. And so on until 4(-) has a 
term in the corner of the half-square. Such is the case for every 
set of repeated roots. All the other terms will be lacking in 4), 
43), .... These results are all obvious consequences of the 
determinantal definition of the coefficients in 7), 3), .... The 
determinant which must not vanish if the solution of the problem 
is possible will reduce to the discriminant of the roots of 2, 
where however the differences which correspond to repetitions of 
the same root have disappeared. This is the only change: for the 
differences which correspond to different repeated roots occur as 
many times as the product of the multiplicities of those roots. 
The proof of these general theorems is carried out by mathe- 
matical induction. It is merely necessary to show that, on the 
assumption that the results are true for any given distribution of 
roots, they still remain true when the number of roots and the 
number of dimensions is increased by one, whether by adding a 
root equal to one already existing or different from all those 
present. In any given case the proof is very simple; but on the 
assumption that there are & roots of multiplicities mi, m2, ..., mx 
the notation becomes very cumbersome. As there is no other 
difficulty than this, it seems hardly worth while to insert the general 
proof at this point. The geometrical consequences of the theorem 
of this article are: That any collineation or strain of which the 
Hamilton-Cayley equation is the equation of lowest degree may be 
converted by multiplication with a reflection /—o|o, which may 
be chosen in co”-1 ways, into a collineation or strain which has 
roots arbitrary except that their product must be the negative of 
the product of the roots of the given collineation or strain. In 
particular these roots may be chosen in such a way that the re- 
sulting collineation or (unimodular) strain may be resolved into 
two reflections. 
19. On the product of a strain and a reflection—Although it is 
evident that no reflection of type 1, nor any reflection of any type 
can be found which will make the scalar invariants of the product 
of any given unimodular strain and that reflection arbitrary, and 
