66 
ON THE THEORY OF DETERMINANTS. 
[12 
Multiplying this by the two sides of the equation 
a, 0, .. 
/3', 
(10), 
and reducing the result by the equation (©), and the equations (6), the second side 
becomes 
k, 0,... 
0, k, 
k, 0 ,0 , 
0, , 
0, !/ (r+1 >, 
(11), 
which is equivalent to 
(r) 
,(r) 
(r+1) ,, lr+i) 
(12), 
or we have the equation 
A 
.. L 
= K r ~ 1 
, V< r > ,... 
jbL (r+l) ) v (r+1) } 
(13), 
which in the particular case of r = n, becomes 
A, B,... 
A', B', 
= K 
•(14), 
which latter equation is given by M. Cauchy in the memoirs already quoted; the 
proof in the “ Exercisesbeing nearly the same with the above one of the more 
general equation (13). The equation (13) itself has been demonstrated by Jacobi 
somewhat less directly. Consider now the function FU, given by the equation (3). 
This may be expanded in the form 
FU = (r£+ st) + ...)[A (Ax + AV +...)+ B (b# + bV + ...) + ...] + (15), 
(r'£ -f s'77 + ...) [A' (Ax + Ex' +...) + B' (b# + bV + ...) + ...] + 
which may be written 
FU = x (A£ + B?7 + ...) + 
x' (A'f + B'rj + ...) + 
(16),