326 Proceedings of the Royal Society of Edinburgh. [Sess. 
and where the S’ s, b’ s, c’s are such that 
8 1 « 1 + S 2 x 2 + 8 3 ^ 3 + S 4 x 4 = 0 , 
b 4 x Y + b 2 x 2 + b z x 5 + b 4 x 4 = 0 , 
c x x 4 + c 2 x 2 + c 3 ^ 3 + c 4 x 4 = 0 . 
The cofactor of m r in | m 1 S 2 b 3 c 4 | being denoted by M r , we see that, as an 
example, 
M 4 2 = 
e f 
b h 
h c 
S 2 8g 
b„ bo 
^3 ^3 
which after performance of the operations 
col 4 - aqco^ - ar 2 col 2 — £ 3 eol 3 , 
becomes 
M, 
row 4 
e 
b 
h 
k 
a?, row, — a; 0 row 0 - x Q vow. 
or say - x?\ 
As it can be shown similarly that 
Mg 2 = - «g 2 A , M 2 2 = - x 2 2 A , M* = - aq 2 A , 
Brioschi obtains * 
I W A^3 C 4 I = ( m i x i + m 2 X 2 + m Z X 3 + n h X d J ~ A J 
nothing being said as to the sign to be taken in extracting the square root 
of x r 2 . 
We have only to add for ourselves that the first of the conditioning 
equations is the vanishing of the quaternary quadric 
/ 9 
h k 
a l 
l d 
and that the <S’s are the halved differential quotients of this with respect to 
rp rp rp rp 
* The minus sign is omitted by him throughout. If the number of cc’s had been odd, 
the sign would have been + . 
