690 
Proceedings of the Royal Society of Edinburgh. [Sess. 
It should be carefully noted also that, while in (2) the number of terms in 
the development is \n(n— 1), in (3) the number is (n — l) 2 . 
Lastly, putting 
0P j 0P 
hiX — ^ 12 ^ — + ' ' 
da rl da rl 
. 7 0P 
da rn 
= H rl 
0P , , 0P , 
OQj r i OCt r 2 
• * + b 2n - — 
da rn 
= H r , 
V 
0P , 0P 
* n m 2 5 
0Ct r i CCt r 2 
d* b nn - 
oa rn 
= H _ 
j 
'so that Hj stands for what P becomes when its r th row is replaced by 
the s th row of Q, and using the multipliers 0Q/06 n , 0Q/06 2 i >••••> dQ/db nl prior 
to addition, Brioschi obtains 
and similarly 
q °P _ , pj i . 
— -n-rlo. + n r2^ — + 
oa rl cb n obey. 
8Q 
0— 1 H 0( ^ 
; 0(X r2 
3 b u 
+ HrS + 
db 22 
+ H ^ 
rn db„o 
Q|^ = H rt ^ + H rt H + 
cCL rn ob ln vb 2n 
With this derived set of equations the multipliers a n , a r2 , . a rn are then 
used, and addition performed, the result being Sylvester’s theorem of 1839, 
namely, 
QP = H rl K rl + H r2 K r2 + • • • • + H TO K m , 
where K rs stands for what Q becomes when its row is replaced by the 
r th row of P* 
In his treatment of the minors of the adjugate determinant Brioschi 
(pp. 36-39) closely follows Spottiswoode ; that is to say, from a set of linear 
equations he derives one result, then from the adjugate set another result, 
and finally draws a deduction from a comparison of the two. His thus 
obtained extension of Spottiswoode’s theorem is open to the same criticism 
as Spottiswoode’s extension of Jacobi’s. 
* Brioschi does not note the independent importance of his second set of equations 
which may be condensed into 
0Q 
Q~ 
OCLrs 
Hr 
0&ls 
+ H<2L + 
do 2 s 
and which, when r, s = l, 1 and n = 3, is 
a l a 2 a 3 
01 02 $3 
7i 72 73 
C 2 C 3 
«1 a 2 a 3 
&1 b 2 &3 
02 ^3 
72 73 
01 02 03 
&i &2 h 
C 1 c 2 C 3 
w 3Q 
00ns 
a 2 a 3 
72 73 
7i 72 73 
&i b 2 & 3 
I C 1 C 2 C 3 
“2 a 3 
02 03 
This, however, may be viewed also as a case of Sylvester’s theorem, namely, where the first 
row of P is 1, 0, 0. 
