Double Products and Strains in Hyperspace. 45 
The collineation 
ee yay nO Ag 05 Ag, O's —Xe-b ay, 0%, — Xs 
which corresponds to the strain 
2—ala-ale+ ele +elyeriytyzitt+dl 
which has been factored into two reflections which in turn corre- 
spond to 
PT 20X% 1 —%1 +-%, OX 2—— %, Oe A hg Og —— hig, 
Sr 0 X14 %};, 2) X p= — Xo -— X3—NX4;, QO XX 3—=-X3 9) Xa; O as 
leaves no non-degenerate quadratic surface Q (4, %,, %3, %4) =O 
invariant. This is clearly seen from the analysis or from the following 
simple geometric reasoning. The collineation (84) has one and 
only one fixed plane f#. This plane must be tangent to the quadric 
Q: for if it cut the quadric in a true conic, the conic being trans- 
formed into itself must have one fixed point, and the plane tangent 
to QO at this point would also be fixed. The fixed plane pf, which 
is tangent to the conic quadric O must intersect the quadric in two 
coincident straight lines or generators: for the collineation (84) can-* 
not have two distinct fixed lines. But the only quadrics which 
can have a double line in common with a plane are the cones or 
other more degenerate quadrics. Hence the theory of collineations 
compounded of two reflections is not quite identical with the theory 
of collineations which leave invariant a non-degenerate quadric but 
includes it. 
On the Resolution of Strains into Reflections. 
17. The product of a unimodular strain by the simplest reflection. 
Consider a dyadic 2 where 2, = + 1, and a reflection /—26| 6 of 
type 1 where the relation 6 6 =1 holds. The scalar invariants 
X55, Xac,---, Xn—1, ¢ Of the-product 
(85) X = 2 (/—2 6/0), 
are determined by the expressions 
(86) Xt s = [Qr (I—2 6 0) |s k= 2, 3,..., n—1 
If (J—2 6|6), be expanded by the binomial theorem, there are only 
two terms in the expansions, namely, 
(87). (J“26\6); Se —2 Le Glo: 
Hence the scalar invariants take the form 
ee 2. (Sr Tay G|6) 9 = Pes oer © (iv Sol) 
It becomes necessary to investigate the expressions Qi ¢ * (Jh1 “ 60) 
more in detail. It is clear that if the scalar quantity 2; - o (Lis 26 6) 
