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

HESSIANS (SALMON, 1852) 391
3a2 = 2a~2 2+a2eAb1 2a 2ab,.-a2 (12 ~ 2a~b~d + b2a3 2 _ aicAb - a1bic2,
3b3 = a3b3 2 + ba2c2 -2b 3blc - b3d2 ~ 2b2cid + CAb12- b2a2c3 - b~c2a3,
3c1 = blec 2 +eib~a3 -2c 1c2a2 -GC1d2 ~ 2c 3a2d ~ ac2 2_- c3b3a1 - c3a3b1,
3a3= b3a3 2 + a3bjc1 - 2a~,a2c2 - a~c12 + 2ac2d + cct2 2 - alb Ic3 - ca1c~b.,
3b1 = C A 2 ~ bic 2a2 - 2b~b3a3 - bid2 ~ 2b2a3d -I- ab32 - b2c2a1 - b act~1
S2 = a22 2 + c2a3b3 - 2c c b1 - c d 2+ 2c b d ~ b c12 - C~a b - C b a,
6d =- 2d 3 + 2d(b1c1 + ca2 +a3b3) + (alb 3c2 ~ b2cla3 + c~a b1) - a b c3
-3(a2b3c, + a,,b1cJ)
The invariants S and T are also printed in full, viz.
S=d4 -2d2 (b~c1+C2a2+a b3) + 3d(a2b3el+ a,,b~c2) - d. a b c3
+ d (a1b3c2 + b2C~a3 + c,,a2b,) - (b~c, C2'L2 + C2a2. a~b3 + a3b3. b~c,)
+ (b1 2c 2 + c22 a2 2+ a32b3 2) - (ajb2. clc2 + b2c2. a2a3 + c~a.. b3b1)
+ (b3c~a2 2 ~ Cictlb 2 + a2b2c1 2 + b2c2a3 2 ~ c3a3b 2 + albi c22),
T -8d6 + 24d4 (bec1+ C2a2 +a3b3) -.
As these differ from Aronhold's by numerical factors, we are
prepared to find corresponding differences in the expressions for
the Hessian of the Hessian and for the discrimin'ant, namely,
4S2. U- T.H(U) and T2 -64S53
respectively.
BRIOSCHI, F. (18,52, August).
[Sur les de'terminants des formes quadratiques. Nouv. Anrnales
de Math., xi. pp. 307-311; or Opere mat. v. pp. 81-85.]
After anl introduction of two pages on determinants in general,
the determinant of a quadratic form is defined as the determinant
whose elements are the second differential-quotients of the form,
the editor adding in a footnote the words, "c'est le determinant
hessierb des Anglais." Starting then from the known fact that
if a~a2 - =b0
a~x, + a2r + 2blXIX2 a, (aix1 + bjX2)2