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

AXISYMMETRIC DETERMINANTS (HESSE, 1853) 13

13.3

"Mlultiplicirt man diese Determinanite nochmals mit der vorhergehenden
und setzt:
F,2 =X1PF'(x,,)+ ~Xl~pF'(X2q) +.+ xnpF'(xQnq),
so erhidt man
l11 V112 ~ ln X11 X21. X1 2 F,, F12... Fill
lt22 l22,. f2m X12 X22... Xn2 _ F21 F22,.. F2,,
'Un1 It... nn X19z X~nx-n F,,1 Fn,2 Fnn.
Da aber F Pq = F22) ist, so ist die letzte Deta-rminante wieder symmelrisch.-"
BRIOSCHI, F. (1853, July).
[Sur uine proprie'ty d'un produit de facteurs line'aires. Gatmbrdgc
and Dub. Math. Jo urn., ix. pp. 137-144; or Opere mat., v.
pp. 121-128.1
This paper of Brioschi's is avowedly an outcome of Cayley's
"On the rationalisation of certain algebraical equations," published
two months earlier. Starting with ' the equation x + y + z = 0,
and using on it the multipliers 1, yz, zx, xy in succession, he
obtains
xyz~+ +Z2. y +y2. Z=O0
Xyz + Z2. X + X2 = 0
xy Z + y2. X + X2. y = 0)
whence on elimination of xyz, XI, y, z there results
0=2 A say;
1 2 X
I 2 X 2
and as- this equation has -its origin in the equation x + y+z =0,
he concludes that the determinant A must have + Y+ z for a
factor. Then-, since any one, of the three other equations
x+y-z=0, x-_Y+z=0, -x+Y+Z=()