(98°) 
(99) 
Double Products and Strains in Hyperspace. 51 
BB?) —a(atet+d)ac’ =A—a(actad+cd)aca 
adc 0) Ap pig mse + a(ac+ad-+ cd) BB NAS 
6 (@-+-0—-a) yy ened day 
+ d(a+a+c) 6S + d(ac+ ac+aa)ds 
E2)a(2Qatd) aa } =3)—a(a?+2ad)ac’ 
a(2a+d) aan cpt ie voitay ta(ar+2ad) aa tere? dary 
a(2a+d) yy? OTOP) +a(a-+2ad) py (OOP? 
d(2a+ a) dS -+-d(a®+2a?) 5d 
5°?) —a(Bajaad E38) —a(3a)aa’ 
+ a(8a) p ae 5 hae — ay + a(8a?) 6B pee macy, 
+ a(3a) yy on A —BO + a(3a?) yy" ses Sia aod 
AG aaa egaaed Te ee 
B%Q—alat2zcac,. | E3)—a(ac+c) aa’ | ; 
+a(a aa COP Wiedc+ ayes. ap 
pee c+2ayyy +c(Qact@)yy | 
+e(c+2a) so P2470 + ¢ Qacta) 6c TV 79 
In the first case the condition that the solution desired be im- 
possible is 
(a—l) x+y |[a(a+e+d) —m|x-+(c+d)y [alac-+ad+cd)—p|x+cdy x 
(a—l)y  [a(ate+d)—mly |a(ac+-ad-+cd)—fply y 
(c_)z2 |c(atat+d)—m\z [c(ad+ad--aa) — p|z Zien 
(d—l)w = [d(a+a+c)—m]|w [d(ac+-ac-+-aa)—p|w zw 
Here the last column may be multiplied by /, m, p, and subtracted 
from the first, second, third columns respectively, and thus the /, 
m, p disappear; the w, 2, y of the last three lines may be canceled 
out; the second line may be multiplied by x and substracted from 
the first and then the y in that line disappears. Finally the second 
line may be subtracted from each of the last two. The condition 
reduces to 
(99’) fea e=-a ca 
@, -a(@--c-d4)  a(ac4-aad--ca) 
c—a (c—a)(a+d) (c—a) ad 
d—a (d—a)(a+c) (d—a)cd 
Here it still is clear that even if the factors c—a and d—a are 
stricken out the expression will vanish if @ equals c or d or if ¢ 
and d are equal. As the expression is only of degree five, it must 
be of the form 
(99") P; (a, c, d) = k (c—a)? (d—a)? (c—d). 
Hence P; does not vanish unless two of the roots a, c, d are equal. 
The second case is between the first and third. In this case it 
turns out that the polynomial to which the determinant reduces is 
= 0, 
Sis) i> ©) 
