370 Proceedings of Royal Society of Edinburgh. [s-kss. 
1 (a-c)( 1+^- 2 ) (a-cf^-' + t 3 ) 
1 (a - c)( 1 + rj~‘ 2 ) (a - c ) f (i? -1 + 77“ 3 ) 
1 (a-c)( 1+r 2 ) (a-c^r + O , 
thence into 
(a-cyt* v ~T 3 
and finally into 
-(a-c)rVT 
£ 3 £ 3 + £ £ 2 +l 
rj 3 77 s + 77 rj 1 + 1 
C 3 £ 3 + £ £ 2 +i , 
I £ 2 
{ £ 2 
(£ + ?? + £- £>?£)• 
Brioschi (1854). 
[La teorica dei determinanti, e le sue principali applicazioni, 
yiii + 116 pp., Pavia: French translation by Edouard 
Combescure, ix + 216 pp., Paris, 1856 : German translation 
by Schellbach, vii + 102 pp., Berlin, 1856.] 
Brioschi devotes the 9th section of his text-book (pp. 73-84) 
to “ determinanti delle radici delle equazioni algebriche,” viewing 
the difference-product and its allies as arising when the roots 
of the equation 
xJ 1 + A n _!a; n-1 + A n _zX n ~ 2 + .... + Apr + A 0 =0 
are substituted for x, and the values of A„_j , A n _ 2 , , are to be 
determined from the n equations thus resulting. His proof, 
obtained in this way, that the common denominator of the A’s is 
resolvable into binominal factors is not of consequence. It is 
more important to note that, as an alternative, he proceeds 
“facendo uso di sole proprieta dei determinanti,” obtaining in 
the first place 
1 
1 . . 
. . 1 
a l 
a 2 
. . a n 
«i ~ a 2 ■ 
a 2 a 3 
■ • a«-i 
“ a« 
a" 
1 
2 
tt 2 *• 
. . a n 
2 2 
tt l — ft 2 
2 2 
a 2 ~ a 3 
. . a" 
71-1 
2 
— a 
71 
n— 1 
a i 
n— 1 
a 2 . . 
n- 1 
. a 
n 
71 — 1 n— 1 
a i “ a 2 
71 — 1 71— 1 
a 2 ' a 3 
71— 1 
•* “»-! 
71—1 
— a n 
from which he removes the factors oq - a 2 , a 2 - a 3 , . . . ; then 
