1910-11.] The Theory of Recurrent Determinants. 
309 
in terms of the coefficients, Brioschi begins by saying that the r\ of the 
said functions, being the complete homogeneous function of degree r, is the 
coefficient of 0 r in the product 
(1 + x 1 z + x 1 2 z 2 + . . . )(1 + x 2 z + x 2 2 z 2 + ... ) . . . (1 + X n Z + X n 2 Z 2 + . . . ) 
x.e. 
He thus has 
1 Qr J_ ga 
(1 - X{z)( I - x,,z) . . . (1 - x n z) ’ 0r Say ' 
1 
4>(z) 
= 1 + N^ + N 2 Z 2 + . . . 
and therefore by differentiation 
| V(z) _ ^ + 2^ + 3^ + 
4>(z) i + ^ 1 2 + K 2 2 2 + . 
But having also by a well-known theorem 
.. = X \ + ^2 + 
x n 
1 -X ± Z 1 - X 2 Z 
1 -x„z 
= x 1 + X^Z + X^Z 2 + 
+ x 2 + x 2 2 z + x 2 3 z 2 + 
he deduces 
+ x n + X 2 Z + X 3 Z 2 + 
= s 1 + s 2 z + s 3 z 2 + . . . , 
Sl + S22 + % , 2+ ... . « 1 + 2^ + 3«^. 
1 + + $2^ + ... . 
whence by equating like coefficients of 0 there results 
{<1 = s i. 
= S 2^^1 S 1 > 
3K 3 = S 3 ^2 S l ’ 
= S r +tf 1 S r _ 1 + .... + Kr-l S l • 
Multiplying now by a r _ v a r _ 2 , . . . , a 0 and adding, he has, on using Newton’s 
relations between the a’ s and s’ s, 
a r-i5^i + 2a r _ 2 5$ 2 + • • • • +ra 0 $$ r = — {m r + (r- 1 )<V-iNi + .... + « 1 fc$ r _i} , 
whence comes Wronski’s relation 
a r + a r _ 1 ^+ . . . +a 0 K r = 0; 
and, on solution of the set of relations obtained from this by putting r= 1, 
2, . . . , r, 
N = 
(-l) r 
°1 
a 0 
a 2 
a i 
a 0 . . 
a r _ 1 
a r - 2 
a r _ 3 . . 
. . a 0 
a 
a r - 1 
a r - 2 • • 
. . o x 
