306 Proceedings of the Royal Society of Edinburgh. [Sess. 
Faure (1855, March). 
[Theoreme sur la somme des puissances semblables des racines. Nouv. 
Annales de Math., (1) xiv. pp. 94-97 : or pp. 172-175 of Combes- 
cure’s translation of Brioschi’s Teorica dei Determinanti .] 
Having seen Brioschi’s result regarding s„ Faure takes up the subject 
and succeeds in throwing on it fresh light. His fundamental proposition is 
not connected with the roots of equations at all, being to the effect that if 
<p(x) = C Q £C m + t Y X m X + . 
. . . + c m , 
and 
f(x) = a 0 x n + flqaj" -1 + . 
. . . + a n 
then 
<p(x) -i-f(x) = A 0 x m ~ n + A 1 x m ~ n ~ 
_1 + A 2 x m ~ n ~ 2 + 
c o 
a o 
. . 
c i 
°1 
a o • • 
C 2 
a 2 
a 1 . . 
C r- 1 
a r -i 
Ct'i — 2 • • 
. . a 0 
C r 
a r 
a r-l • . 
. . 
Having stated this he recalls the theorem * that if 
f( x ) = a 0 (x -x l )(z- x 2 ) ... (x - x n ) 
we have 
*A( x ‘i) + + • • • + l l / ( x n) = coeff. of x~ l in / {x)\p(x) 
f\ x ) 
and therefore as a special case 
x Y r + x 2 r + ... + = coeff. of x~ r ~ l in yj-j • 
It is thus seen that to obtain a determinant expression for s r we have only 
to make cp ( x ) identical with/' (x ), — in other words, put m = n— 1, c 0 = na 0 , 
c 1 = (n — l)a v c 2 =(n — 2)a 2 , . . . , and find the coefficient of x ~ r ~ . Doing 
this we obtain 
na Q 
a 0 
(n - 1 )« x 
a x 
a, . . . 
(n-2 )a 2 
a 2 
a i • • • 
(n - r)a T 
a r 
a r-l . • • 
* Said to be first given by Cauchy in his Exercices de Math, for 1826. 
