1907-8.] 
where 
Dr Muir on the Theory of Hessians. 
425 
aj = a Y d 2 + & 1 a 3 2 + cp^ 2 - a-^b^ — 2 da 2 a z . 
b 2 = & 2 d 2 + c 2 b-^ + a 2 5 3 2 - a 2 b 2 c 2 - 2db z b 1 . 
c 3 = c 3 c7 2 + a 3 c 2 2 + b z c^ - a z b z c z - 2dc 1 c 2 . 
3a 2 = c 2 a 2 2 + a 2 c 1 b l - 2a 2 a z b z — a 2 d 2 + 2a x b z d + b 2 a z 2 - « 1 c 1 & 2 - cq&jCg . 
3b g = a 3 6 3 2 + b z a 2 c 2 - 2b z b 1 c 1 - b z d 2 + 2b 2 c 1 d + c z b - 5 2 a 2 c 3 ~~ ^ 2 c 2 a 3 • 
3(q — + c-\b z a z - 2 CjCgttg - c x d 2 + %e z a 2 d + a x c£ — cjb z a x - c z a z b 1 . 
3a 3 = b z a z 2 + a z b 1 c 1 -2a z a 2 c 2 - a z d 2 + 2ci 1 c 2 d-\-c z a 2 2 - a 1 c l b z . 
3bi = c-fi^ 2 + b^c 2 a 2 - 2b 1 b z a z — b-^.l 2 + 2 b 2 a z d + aq& 3 2 — . 
3c 2 = a 2 c 2 2 + c 2 a 3 & 3 — 2c 2 c 1 & 1 - c 2 c^ 2 + 26' 3 5 1 ^ + 5 2 Cj 2 — c 3 a 3 & 2 - c 3 6 3 (X 2 . 
6d = - 2d 3 + 2d(b l c 1 + c 2 a 2 + a z b z ) + (aq&gCg + b 2 c Y a z + c 3 a 2 & r ) - ajb^c 3 
- 3 (a 2 6 3 c 1 + ag^Cg) 
The invariants S and T are also printed in full, viz. 
8 i # - 2d 2 (b 1 c l + c 2 a 2 + a z b z ) + 3 d(a 2 b z c 1 + a 3 %|) - d-a 1 b 2 c z 
+ d(a 1 b z c 2 + b 2 c x a z + c z a 2 b j) - (^rq • c 2 a 2 + c 2 a 2 • « 3 & 3 + a 3 5 3 • b^) 
+ (b 2 c 2 + o 2 Gj 2 H- a 2 b 2 ) — (oq b 2 • tqc 2 + & 2 c 3 • ^ 2 ft 3 + c 3 oq • frg^q) 
+ (b z c z a 2 2 + c 1 a 1 5 3 2 + ajb 2 c-^ + b 2 c 2 ci z 2 + c z a z b-f + a 1 5 1 c 2 2 ) , 
T = - 8d 6 + 24ri 4 (& 1 c 1 + c 2 a 2 +a 3 & 3 ) - 
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 discriminant, namely, 
4S 2 -U - T-H(TJ) and T 2 -64S 3 
respectively. 
Brioschi, Fr. (1852, August). 
[Sur les determinants des formes quadratiques. Nouv. Annates de 
Math., xi. pp. 307-311.] 
After an 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 hessien des Anglais.” Start- 
ing then from the known fact that if a ± a 2 — 5 X 2 = 0 
1 2 
flq.«q 2 + « 2 aq 2 + 2b l x 1 x 2 = - (aqaq + b Y x^) , 
a i 
Brioschi states that similarly, if the determinants of 
flqaq 2 + a 2 x 2 2 + a s x z 2 + 2b 1 x l x 2 + 2b 2 x ] x z + 2 c 1 x. 2 x z , 
aqaq 2 + a 2 x . 2 2 + 2 b l x l x 2 , 
flqaq 2 + a z x 3 2 ■+■ 2b 2 x 1 x z , 
