ON THE THEORY OF DETERMINANTS. 
69 
12] 
which demonstrates the equation (34); and this equation might be proved in like 
manner from the second of the equations (33). If however, J = 0, or / = 0, the above 
proof fails, and if KU = 0, the proof also fails, unless at the same line n = 2. In all 
these cases probably, certainly in the case of KU=0, n 4= 2, the equation (34) is not 
a necessary consequence of (33). In fact FU, or DU may be given, and yet U 
remain indeterminate. 
Let U /} a,, /3,, ...A„ B,, &c.... be analogous to U, a, /3..., A,, B, &c.... and 
consider the equation 
K(KU,.FU+gKU.FU,).. 
tc,A + g/cA,, k,B + gtcB,, ... 
k,A' + g/cAf, kB + gK 5/, 
(40), 
Multiply the two sides by the two sides of the equation (2), the second side 
becomes, after reduction, 
Multiplying by the two sides of the analogous equation 
*,= «,» «/> 
£,» ft, 
(42). 
and reducing, the second side becomes 
KK, (o, + go. ), KK, (/3, + g13 ), 
KK , «+9 a ')> KK , (£/+gfi')> 
whence 
= ic n .K, n .K(U, + gU) 
K {KU,. FU+gKU. FU,) = (KU)"- 1 (KUy-'K(U, +gU) 
(44) , 
(45) , 
and similarly 
o% + /377..., a, + go ..., /3, +g/3 ..., 
a'f + iSV.., af+got..:, £/+flÆ/-.