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74 
ON THE THEORY OF DETERMINANTS. 
[12 
whenever s is less than the number expressing the order of the determinant. Hence 
in the formula (70), if S contain a single term only, the first side of the equation 
is linear in p, a, ... and also in a, a', ..., i.e. it consists of a term independent of 
all these quantities, and a second term linear in the products pa, pa', ... a a, era', ... 
This is therefore the form of K (U + U). 
Consider the several equations 
k = KTJ=Aol +B/3 + 
= AW + B'/3' + ... 
= &c. 
it is easy to deduce 
k=K(U+ U) = KU + A pa + B aa + 
+ A'pa' + B'aa + 
To find the values of A, B, &c. corresponding to U + U, we must write 
A = m' /3 + n' 7' + 
= m"/3 + n V + 
= &c. 
where 
(72), 
(73). 
•(74), 
m = 
m = + 
// 
7 > 
h",... 
+i 
II 
V; 
S", 
// 
e , ... 
/// 
7 > 
8”', 
8'", 
/// 
e , 
/// 
7 > 
8'",... 
, N" = 
8'" , 
e'",... 
//// 
7 > 
8"", 
8"", 
//// 
e , 
(75), 
, &c. 
the order of each of these determinants being n - 2, and the upper or lower signs 
being used according as n — 1 is odd or even, i.e. as n is even or odd. Hence 
A / = A + m' aa' + n' to! + (76), 
+ m" aa!' + n" to!' + ... 
and therefore 
K t A — K.A t = A 2 pa + (AB ) aa + (AC )ra + ... 
+ A A' pa! + (-AB' — /cm' ) aa' + (AC — kn' ) ra! + ... 
+ A A" pa" + (AB" - km") aa" + (AG" - *n") ra" + 
•(77). 
the additional quantities G, r having been introduced for greater clearness. Now the 
equations 
AB' - kM' =A' B, AC-kN' =A'C, (78), 
AB" - kM" = A"B, AG" - kN" = A"G,