Double Products and Strains in Hyperspace. 49 
The converse problem is interesting, namely, given a dyadic Q, 
to find what dyadics may result from the product 2 (J—2 6|o) by 
a suitable choice of the reflection 7—2.6\6. Consider the scalar 
invariants of the baa a The determinant of the product must be 
the negative of 2,. Suppose it be desired to make the other scalar 
invariants take assigned arbitrary values. This amounts to the so- 
lution of the equations 
(96) G26=/ G2 iho 
6 226m or =o (2) —mI)o=—0 n—1 equations 
6 EB G=p o(z Foal eee 
under the condition 66-0. e ike: tore it amounts to finding 
a space o which shall contain (Q —/J)o, (Z?)—m/)o,... but not 
contain ¢. Suppose the roots of 2 are distinct and for simplicity 
letin—4. Then 
Q=aala +bBlB+ervly +dolv 
(96) B-—a(b+c+d)alad+6 (e+d+a) B\B +c(d+a+d)yly 
| @ (2-0-6) 0/0" 
Be) —a(bc + bd+ cd) ale +b(cd+ca+da) B|p 
+c(da+db+ab)y\y +d(ab+ac-+ bc) 8|6 
o=—xatyB+yy+wo 
The three vectors (Q—-/1)o, (5@)—mT)o, (3@) —pJ)o are easily 
written down and the desired 6 may be passed through them un- 
less the condition 
(97) |(a—/) x [a(6+c+d)—m]«x |a(bc+bd+cd)—p|x x| 
\(6—A)y [b(c+d+a)—mly [b(cd+ca + da) ~ Al: yoy 
(c—l)z [c(d+a+6)—m]z2 [c(da+db+ab)—p|2 
(d—/)w [d(a+ 6+ c)—m|w |d(ab+ac+bc)—flw w 
which expresses the fact that they lie in a 3-dimensional space 
with o, holds. This may be reduced to the simpler form 
Pamia a(o--¢c4-d) a(be+-bd+eca) 1) 
16 b(c+b+a) b(cd+catda) 1|_ 
c e(dta+b) e(dat+db+ab) 1 Fi (a, 6,¢,d)=0 
Gaia 04-0 )s did 4- a6 4 pe) 1) 
This is a polynomial of degree 6 in a, 6, ¢, d. It is obvious that 
the polynomial will vanish if any two of the roots are equal. Hence 
(97") P,(a, 6, c,d) = k (a—b) (a—c) (a—d) (b—c) (6—d) (c—a), 
and according to the supposition that the roots are distinct P; == 0. 
From this it follows that if the roots of 2 are distinct, a reflection, 
I—26\6 of type 7 may be found which will make the product 
2 (1—2 o\6) take such a form as to have any desired scalar in- 
variants, with the exception of the determinant which is —2, 
Trans. Conn. Acap., Vol. XIV. 4 SEPTEMBER, 1908, 
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