12] 
ON THE THEORY OF DETERMINANTS. 
71 
then in (45) taking g = -1 [the Numbers (56) &c. which follow are as in the original 
memoir] 
K{K{U+U). FU- KU. F (U + U)} = K {K (U + U). UU- KTJ. V (U+ U)} (56), 
= (KUy-' .(K(U+ U)Y~ 1 . ku, 
or observing the equation (50), 
K{K(U+ U).FU-KU.F(U+U)} = K {K(U + U).VU - KU .V (U+U)} = 0 (57). 
Hence F{(K(U+U). VU-KU.V {U + U)} = U [K (U + U). FU - KU. F (U+ U)\ 
are each of them the product of two determinants. But this result admits of a further 
reduction: we have 
F{K(U+ U). UU-KU. V (U+U)} = U{K(U + U).FU-KU.F(U + U)} (58) 
= - // (.KU) n ~ 2 . {K (U+ U)) n ~ 2 
apAr a'x'+ ..., fi t x + fi'x' + ... , ... 
«Z + fiv + •••, , /3, -/3 
«'£+/3^ + ..., a/ - a' , /3/ + fi' 
substituting a / = a -f Spa, &c...., also observing that if the second line be multiplied 
by x, the third by x\ ... and the sum subtracted from the first line, the value of 
the determinant is not altered, and that the effect of this is simply to change 
a,, a/ ... into a, a ... in the first line, and introduce into the corner place a quantity 
— U, which in the expansion of the determinant is multiplied by zero: this may be 
written in the form 
-Jr{KU) n ~*(K(U+ U)) n ~ 2 
ax + a'x' + ..., 
fix + fi'x' + ..., ... 
+ 0V + ..., 
Spa 
Sera , 
a'Z + fi' v + ..., 
Spa' 
Sera' , 
which may be reduced to 
Jf. (KU) n ~ 2 .(K(U+ U)) n ~ 2 X 
ax + a'x' + ..., fix + fi'x' + ..., ... 
+ fiv + ■.. 
, a'^ + fi'v 
P 
O’ , 
a 
a' 
(59), 
(60), 
If each of these determinants are multiplied by the quantity (.KU) n ~ l , expressed 
under the two forms 
A, B,... 
A, A',... 
A', B', 
> 
B, B', 
(61),