153 
L 906— 7.] Dr Muir on Axisymmetric Determinants, 
so erhalt man 
u n 
U 12 • ' 
■ • u ln 
*11 
*21 * 
■ • *«1 
2 
Fa 
-^12 • ‘ 
. i\„ 
U 21 
u 22 . . 
• • U 2n 
. 
*12 
^22 * 
• • *n2 
= 
F 2 i 
-^22 • * 
• I'm 
U nl 
u n 2 • • 
• • Unn 
x m 
*2n \ ■ 
• • X nn 
F nl 
N n2 . 
■ ■ 
Da aber F P5 = F qp ist, so ist die letzte Determinante wieder symmetrisch.” 
Brioschi (1853, July). 
[Sur une propriety dun produit de facteurs lineaires. Cambridge and 
Dub. Math. Journ., ix. pp. 137-144: or Opere Mat.] 
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 + 0 = 0 , and using on it the multi- 
pliers 1 , yz , zx , xy in succession, he obtains 
x + y + z = 0" 
xyz + + z 2 .y -f y 2 .z = 0 
xyz + z 2 .x + x 2 .z = 0 
xyz + y 2 .x + x 2 .y = 0 
whence on elimination of xyz ,x, y , z there results 
0 = 
.111 
1 . z 2 y 2 
1 Z 2 . X 2 
1 y 2 x 2 . 
= A say ; 
and as this equation has its origin in the equation x+y + 2 = 0 , he con- 
cludes that the determinant A must have x-\-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 = 0 
gives rise to the same result, it is easily seen how he reaches the identity 
.111 
1 . z 2 y 2 
1 z 2 . x 2 
1 y 2 x 2 . 
(x + y + z)(x + y- z)(x -y + z)(-x + y + z). 
An alternative form of A is introduced by the words “ Observons qu’on 
a evidemment * 
* It would seem preferable to multiply the columns of A in order by xyz , x , y , z , the 
quantities just eliminated ; and then divide the rows by the multipliers 1 , yz , zx, xy used 
in obtaining the set of equations. The advantage of this method would be still greater in 
the next case. 
