Double Products and Strains in Hyperspace. 44 
| a's + Bl B's teyily's +..., &+-14 S25, and so forth, does 
not effect the nullities of Z, Z°, Z°, ... and hence does not alter 
the type of the dyadic. Hence if the transformation which carries 
(79") into (79') be obtained and applied to the factors (81), the 
factors will take the desired form. 
From the preceding analysis it is seen that in every case either 2 
may be factored or may be referred to similar transformations in 
a less number of dimensions. As the proof that any dyadic which 
satisfies a reciprocal equation has been given when = 2 and when 
n= 3,1 it follows that: The necessary and sufficient condition that 
a dyadic # be factorable into two square roots of the idemfactor 
(or geometrically, that a strain be resoluble into two oblique re- 
flections) is that the scalar invariants 
Ds, Dos, ..- , On—1,5s, On = mr 
be such that the scalar equation is reciprocal and that the sets of 
invariant numbers 
By Rises 5 Mp2; heat 
which correspond to any root and its reciprocal be equal. This is 
the generalisation of the result I obtained for the case n= 3. 
16. Kelations between strains and collineations.—If a strain in x 
dimensions be written as a matrix by chosing the antecedents and 
consequents as reciprocal systems, so that 
(82) 2= Cy |e +1 a \e'o +... + Cindy |a'n 
C91 Gg Oy + Co Gg|a'g +... + Con Go| @'n 
+ Cni Gn | a's + Cnan|a'g +... + Cnn an| en 
and Deepa Lo Oo cite ais an On 
the transformation @ = 2 in oblique coordinates becomes 
(82’) My = Ceti Cigxa +... teint 
Kg = Co, %1 + Coy Xo +... + Con Xn 
GEA =n 1 — Cng Xo a musts + Cnn Xn 



1 It should be noted that in these cases the additional condition that 
the invariant numbers be equal for reciprocal roots is fulfilled necessarily 
as the equation of lowest degree completely characterizes a dyadi¢ when 
na<4. The treatment for x= 3 is given in the first reference of p. 32): 
the treatment for = 2 may be regarded as a special case of that or as 
a special case of the investigation I gave in A generalized conception 
of area: applications to collineations in the plane--The Annals of Mathe- 
matics, second series, volume 5, pp. 29—45 (1903). 
