307 
1910-11.] The Theory of Recurrent Determinants. 
which is readily reduced to 
«1 
a 0 
2 a 2 
a i a 0 . . 
ra 
a r _ i a r _ 2 . . 
. . 
and so agrees with the result obtained from Newton’s relations between 
the as and s’s. 
Here Faure leaves the subject, but he might equally easily have 
established Brioschi’s more general result. Instead of specialising by 
putting \Js(x) = x r he might have made \Jy(x) = x r (p(x)f\x) and so have got 
x 2 r cf>(x 2 ) x n '<fi(x n ) _ coe ^ x -r-i j n ^0*0 < 
f\ x i) f(x 2 ) ” " f(x n ) * f(x) ’ 
Bearing in mind that <p(x) as used by Brioschi was of the ^ th degree, we 
have from Faure’s fundamental theorem the said coefficient 
=A r+1 =(-ir i 4 
a 0 
c 0 
a 0 
C 1 
a l 
a o 
C r 
a r 
a r -i 
a 0 
c r+ 1 
a r+ 1 
a r 
Oj 
as it ought to be. 
Bruno, 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. e Fis., vi. pp. 479-480. 
The formula referred to is 
0^+i 
0^+i 
^{H x )} = 
\j/ mf/ '<f> 
ln{n- l)i//"</> . 
. . . 
- 1 ip'p 
(n- 1)0 
xj/ {n) <P 
-1 
xf/'P 
. . . \p (n ~ x) <f> 
- 1 
. . . if/ (n ~ 2) xf/ 
\p'<p 
where the coefficients in the r th row are those of the expansion of 
(a + b) n ~ r+1 , and where after development of the determinant <p r is to be 
taken as meaning the r th differential-quotient of (p with respect to \Js * 
* 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. 
