12] 
ON THE THEORY OF DETERMINANTS. 
65 
Let A, B, ... , A', B',...,... be given by the equations: 
A = 
, B=± 
i, S',... 
r, y"> 
// £// 
7 » o , 
A'=± 
Qtt " 
P > 7 > ••• 
, B' = 
7,6,... 
/D'/' /// 
P > 7 > 
7", S'", 
(The upper or lower signs according as n is odd or even.) 
These quantities satisfy the double series of equations, 
Act + Bf3 -f ... = k, (6). 
A a' +B/3' + ... =0, 
A'a+B'P + ... = 0, 
AW + B'/3' + ...=*, 
&c. 
Act + AW +... = k, (7), 
Ap + A'P+... = 0, 
Bet + BW + ... = 0, 
B&+ B?P + . .. = *, 
&c. 
the second side of each equation being 0, except for the r th equation of the r th set 
of equations in the systems. 
Let A, fj,,... represent the r th , r + 1 th , ... terms of the series a, /3,... ; L, M,... the 
corresponding terms of the series A, B..., where r is any number less than n, and 
consider the determinant 
A ,...L 
(8). 
A^-v,... L (r-1) 
which may be expressed as a determinant of the w th order, in the form 
A ,...L ,0, 0,... 
A^- l \ , 0, 0, 
0 , 0 , 1, 0, 
0 , 0 , 0, 1, 
(9).