12] 
written under the form 
ON THE THEORY OF DETERMINANTS. 
AB' - A'B = kM', AC -A'C =kN\ ... 
AB" - A"B = kM", A G" -A"G = kN", 
are particular cases of the equation (13), and are therefore identically true. Hence, 
substituting in (77), 
K t A — K.A t = A 2 pa + A Baa + A Gra ... + 
+ A A'pa' + A'Baa'+ A'Gra ... + 
+ A"Apa" + A"Baa" + A"Gra" ... + 
= (pA + aB + ...) (.Aa + A'a' + ...). 
Forming in a similar manner, the combinations k,B — kB,, ... K t A' — kA'^ K t B — k,B' , ..., 
multiplying by the products of the different quantities Ax+A'x'..., Bx + B'x'..., ... 
Ri- + Sr)..., R'i; + S'rj..., ... and adding so as to form the function K (U + U). FU 
- KTJ . F(U + U), we obtain the required formula, viz. that the value of this quantity is 
= [(pA + aB ...)(Bf + S v ...) + (A'p+B'a ...) (r'^+s't; ...) + ...] (81); 
x [(Ha + iV ...) (Ax + aV ...) + (Ba + B'a'...) (b# + bV ...) + ...] 
with this theorem, I conclude the present section,—noticing only, as a problem worthy 
of investigation, the discovery of the forms of the second sides of the equations 
(G8), (69), in the case of S containing more than a single term. 
§ 2. On the notation and properties of certain functions resolvable into a series 
of determinants. 
Let the letters r 1} r 2 , ...r k 
represent a permutation of the numbers 
1, 2, ... k 
Then in the series (1), if one of the letters succeeds mediately or immediately 
a letter representing a higher number than its own, for each time that this happens 
there is said to be a “derangement” or “inversion.” It is to be remarked that if 
any letter succeed s letters representing higher numbers, this is reckoned for the 
same number s of inversions. 
Suppose next the symbol 
+ . 
denotes the sign + or —, according as the number of inversions in the series (1) is 
even or odd. 
10—2