The theory of determinants in the historical order of development, by Sir Thomas Muir.

214 HISTORY OF THE THEORY OF DETERMINANTS

Here Faure leaves the subject, but he might equally easily have
established Brioschi's more general result. Instead of specialising by putting +f(x)=xr he might have made +(x)=x '(x)f'(x)
and so have got
x'(x) + 2(X2) + + X.T. + (Xn) = coeff. of x-,-l in f(X)
/'l) /'(X) f '(n)
Bearing in mind that +(x) as used by Brioschi was of the nth
degree, we have from Faure's fundamental theorem the said
coefficient
C1 C11 0
= A.+ = (- 1)+ 1 c a0......
Cr a,1 a-....
C, I, a,+,.(t, C1
Cap d- Ct,r+i C1r..... 61
as it ought to be.
BRUNO, F. FAA DI (1855, December).
[Note sur une nouvelle formule du calcul differentiel. Quart.
Journ. of Math., i. pp. 359-360; or, with a different title,
Annali di Sci. mat. efis., vi. pp. 479-480.]
The formula referred to is
an+l V- n= _ 1 /('-1)` (l, '"~ V(2+1)%
- -1 j'~ (1.... )(X. -1..-I.... I..... V'V
where the coefficients in the rth row are those of the expansion
of (a+b) -2r+1, and where after development of the determinant
(r is to be taken as meaning the rth differential-quotient of <
with respect to ~.r
* An opportunity was here lost by Bruno of noting that a recurrent with the
elements in its zero-bordered diagonal all negative has all its terms positive.