12] 
ON THE THEORY OF DETERMINANTS. 
79 
I proceed to prove a theorem, which may be expressed as follows: 
+ 
t 
where 
(28), 
(27) 
the number of the symbols r, s,... being obviously 2p — 1, and that of x, y,... being 
2q — 1. The summatory sign S refers to l, and denotes the sum of the several terms 
corresponding to values of l from l = 1 to l = k. Also the theorem would be equally 
true if l had been placed in any position whatever in the series r, s ...l; and again, 
in any position whatever in the series x, y... I, instead of at the end of each of these. 
With a very slight modification this may be made to suit the case of an odd number 
instead of one of the two even numbers 2p, 2q; (in fact, it is only necessary to place 
the mark (T) in [Ah\ ...} over the column corresponding to the marked column in 
\A ...}, [A ...} being the symbol for which the number of columns is odd), but it is 
inapplicable where the two numbers are odd. Consider the second side of (27); this 
may be expanded in the form 
2 + ¿a • • • ix iy • • • Ah j g 1 ,..x L y 1 ... • Ah 2$2...... • • • Ah|fcs k ...x k y k ... (29), 
where 2 refers to the different quantities s, ..., x, y,... as in (11). 
Substituting from (28), this becomes 
2. ... Si k (+ ±«... ± x ± y A 1Sl '"i i ... A kSk '"i k ... iW.A • 
Effecting the summation with respect to x, y... this becomes 
t 
2 now referring to s, ... only. The quantity under the sign 2 vanishes if any two 
of the quantities l are equal, and in the contrary case, we have 
which reduces the above to 
t 
[h. k. 2q). 2 + ± s ... ±iAi'g l ..'i l ... A k . 
(33), 
2 referring to the quantities s..., and also to the quantities l. And this is evidently 
equivalent to 
+ 
t 
{A . k . 2p\ [h. Ic. 2q) 
(34), 
the theorem to be proved. It is obvious that when p= 1, q=l, the equation (27), 
coincides with the theorem (©), quoted in the introduction to this paper.