Double Products and Strains in Hyperspace. 35 
the product. Which of the spaces Vy4i-2m and Vy_-x-i42m is 
identically fixed and which is involutorily fixed depends on the 
sign of 6 #. If the sign is +, V,—x-142m is identically fixed. The 
problem of determining the conditions under which two square 
roots of J are homologous may therefore be considered as solved. 
15. The product of two involutory transformations.—Next consider 
the product of any two involutory transformation, 2 = # #@, where 
k = 
(69) f= + F--2> a;| a) pes EG 3) 
fs 1 
l ms 
p— + (—2 WN £;i| B:) es zt 
1 ~_ 
As @ and ¥ are their own reciprocals, 2-1! = @ # by (14). Hence 
by (35) 
(70) Pe i Cor Pe — re 
On substitution from the relations (30), there results 
(70') Qrs = Qn Qn—h, s, Qn = ats ile 
There arise, then, four different cases of the scalar or characteristic 
equation (40): 
(ia) x" — 2. Kn Sas X27 —...— 22 4° Ds x -1=0; nodd, 2, =0, 
Oe ng a — ot eg 8 ee t+ — 0, meveny i, 0) 
c—-2,4" 1 Oo, x2 —...+0),%° 2, *£+1=0, nodd, &, —0, 
4°— QD, X91 Dog X"—2—...— Do 5 x7 + 2s X-1=0, neven, 2, —0, 
according as z is odd or even and 2 proper or improper. The 
first three of these equations are reciprocal equations, the last is 
not, unless Qns = 0. Thus, if a dyadic can be written as the pro- 
2 
duct of two square roots of the idemfactor, the scalar equation is 
reciprocal except in the case that z is even and the determinant of 
the dyadic is negative. This case is treated later. 
If the number of dimensions is odd, the determinant of —/ is 
negative. Hence the third case in the above list may be reduced 
to the first case by making the simple change of 2 to —2. More- 
over, if the question of interest were to decide whether, given a 
reciprocal scalar equation, every dyadic which satisfied it were 
resoluble into two reflections, it would be sufficient to answer the 
question for dyadics of positive determinant, in case 7 is odd, in- 
asmuch as —/ is commutative with any dyadic. In the fourth case, 
it would be possible to replace 2 by 2 (J—ea\a) or by 2& times 
any reflection of Saipan: —1. But here nothing is gained, 
because the product 2 (J—«\«) may not satisfy an equation of the 
