Double Products and Strains in Hyperspace. 33 
respectively the idemfactor and its negative; and the other types 
occur in pairs, namely, 1 and wm—1, 2 and m—2,... which differ 
only by the factor —1.. The number of square roots of type & is 
co2k(n—k). Another form in which #@ may be expressed is 
ke 3 
(68) $’—-+ 1—2za|ai), RS (2) 
1 _ 
n\ . é n 
where (5) is the integral part of 5 and where the relations 
(68) a a=1, «a; a; =0 
hold. For some purposes this form is more convenient. It should 
be remarked that what is important is not the individual antecedents 
and the individual consequents, but the spaces 
ABE, Gey. 12),° Rx) ke oie (cy, i ae ee eh) 
of & and of z—& dimensions which are determined by them. It is 
clear that the spaces AR, and Rp; are invariant under the trans- 
formation #; the former having all vectors identically fixed and 
the latter having all vectors reversed in direction or vice versa, 
according as the — or the + sign is taken with the parenthesis. 
To consider the transformation of vectors in general, it will be 
best to resolve the vectors along the two fixed spaces. Then the 
component along the space identically fixed will remain fixed, and 
the other component will be reversed in direction. It is clear that 
if either of the fixed spaces be taken with all the dimensions of the 
other fixed space except one, the result be a space of w—1 di- 
mensions which will be fixed. The volume of an 7-dimensional region 
is not changed in magnitude or in sign by the even types 0, 2,...; 
and is changed only in sign by the odd types 1, 3... As the 
transformations may evidently be regarded as a generalisation of 
reflection, namely a reflection through the space Ry (a, Go, ... @x) 
parallel to the space Ry; (a1, a, ..., @k) Or vice versa, according 
as the — or + sign is used, the designation ‘oblique reflection’ or 
merely ‘reflection’ will be applied to the geometric counterpart of 
the square roots of the idemfactor. In case the volume does not 
change sign the reflection will be called proper, in other cases it 
will be called improper. And these terms will be used to apply to 
dyadics in general; if #, > 0, the dyadic is a proper dyadic, and 
if Pp < O, it is improper. 
If two square roots of the idemfactor are to be homologous, 
they must be commutative. It is a general theorem in transfor- 
mations that the necessary and sufficient condition that the product 
of two involutory transformations be commutative, is that it shall 
itself be involutory. Hence two square roots of J will be hom- 
Trans. Conn. Acap., Vol. XIV. 3 SEPTEMBER, 1908. 
