Double Products and Strains in Hyperspace. 55 
The connection of the conditions (102°) with work which has 
already been accomplished is this. In case 7 — 2, the only reflection 
is of type 1 and if 2,——1, X,—-+ 1, there is no condition to be 
fulfilled. Hence the transformation 2 may always be written as 
the product of three reflections. If 2,—-+-1, the Hamilton-Cayley 
equation of 2 is necessarily reciprocal and ® is resoluble into two 
reflections. These results are well known; and as far as collineations 
are concerned there is no distinction between the two cases except 
as regards reality. If #—283, there is only the one reflection, which 
is of type one, (except for its negative) and only one condition — 
which may always be satisfied... Hence in three dimensions the 
resolution into three reflections is always possible. In case n—4 
and @ is of type 1 and 2,——1, there is still only one condition 
(102°) to be satisfied, and it can clearly be satisfied: but another 
: : . ; 1 
difficulty arises owing to the fact that if a, a are double roots of 
the product X, it may conceivably arise that for all reflections @ 
which satisfy the condition there may be a shearing term for one 
of the roots and none for the other, so that the supplementary 
condition concerning the invariant numbers would not be fulfilled. 
In a delicate question of this sort a count of constants is of no 
value; a detailed investigation of the product X is required. 
Whereas if 2, —-1, there are two conditions (102’) to be satisfied 
simultaneously, and in view of the developments of article 47 it is 
by no means evident that this may always be accomplished. If ® 
is of type 2 and 2,——1, there are again two conditions (102) 
to satisfy, not to mention the conditions imposed by the invariant 
numbers, and again it is not obvious that they can be met. If 
however 2,-—-+1, Smith’s theorem previously cited, and arising 
out of the special fact that a collineation in four homogeneous 
variables may be regarded as a collineation in six variables with 
an invariant quadratic form, may be adduced to show that all the 
conditions (102) may be satisfied. If > 4, the difficulties signalised 
for the first three cases when 2—4, are further emphasized. 
To show that these difficulties are not only conceivable but ac- 
tually arise, it is worth while to treat the simplest case. Suppose 
(103) 2=aala'+aé@\e+ayly +a36|6', Q9=4+1, O=1—2 ol0 
Here there are two conditions so satisfy, namely 
(104) o[2— 2-'— }$(2, — 2") I]o=0, ofF@ —423,1|o=0. 

‘ This is precisely the condition of my theorem 23, p. 295, of my 
communication to the Transactions cited in the footnote on page 32 
