The theory of determinants in the historical order of development, by Sir Thomas Muir.

AXISYMMETRIC DETERIMINANTS (CAYLEY, 1846.)13

113

A-xa H-xh G-xg
H-xh B -xb F -xf =0
G-xg F-xf C-xc
has all its roots real if either of the quadrics
AX2 + By2 + CZ2 + 2Fyz + 2Gzx ~ 2Hxy,
ax2 + by 2 + cz2 + 2fyz + 2gzce + 2hxy,
remain constant as to sign.
CAYLEY, A. (1846).
[Proble'me de geometrie analytique. Crelle's Journal, xxxi.pp. 227-230; or Collected Math. Papers, i. pp. 329-331.]
The problem in question depends on an algebraic identity
which, after a little examination, is seen to be a property of
axisymmetric determinants. Cayley writes the identity in the
form
F19V (U~Y 2) K (U) - p F~(U).K (U + V2) = O()
where
U -AX2+ By2~+ CZ2 + 2Fyz ~ 2Gzx + 2Hxy + 2Lxw + 2Myw + 2Nzw +Pw2,
V= a(X+iy+Yz~8w,

ARH
H B
G F
L M

G L
F Al
C N
N P

e ' Cto
a A H GL
P,,,(U) = j3 H B F M
G F CN
8 LM-qN P

and P,,(U) is what is obtained from P,,(U) on changing a, ~3, y,
8 into e, n, ~, w respectively: but freed of all fresh notation it is
nothing more nor less than

$ A+a 2
jq H+/3a
G+y/a
w L +da
M.D. IL.

I A
H + a3 G~ay L +a8 '
H~~

H G
B F
F C
M N

L
Ml
N
P