86 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 



which can be formed out of the numbers (2). The equation (11) would stiU be true, if the 

 mark (f) were placed over any number of the columns p, a... . 



Suppose in this equation a single column only is left without the mark (f) on the second 

 side of the equation ; the first side is then expressed as the sum of a number 



(1.2... A;)"-', or generally (1 . 2 . . . A:)"-'-', (13), 



of determinants, according as we consider the symbol (4) or the more general one (8). And 

 this may be done in (w) or n-.v different ways respectively. 



It may be remarked, that the symbol (8) is the same in form as if a single column only 

 had the mark (f) over it; the number («) being at the same time reduced from (n) to (n - a? + 1). 

 For, the marked columns of symbols may be replaced by a single marked column of new symbols. 

 Hence, without loss of generality, the theorems which follow may be stated with reference to a single 

 marked column only. 

 Suppose the letters 



jO,, pi.-.p/,'- <^i^ <r2...CTt; &c (U) 



denote certain permutations of 



a„a,...a^; /3,, /3, ... /B*; &c (15), 



in such a manner that 



/>i = a,_, p2 = ci^^.. pt = a^^; cr, = /3i , 0-2 = /3j^, ..0-4= ^A^; (ifi)- 



Then the two following theorems may be proved : 



U,o,<t,..(m) = ±,±A.- Ua,/3,..(w)| (17). 



If (w) be even, but in the contrary case 



(J|0,<r,..(«)| = +±p-- Uai/3,..(w)l (18). 



Pk<^k ' '■ "kPk 



By means of these, and the equation (li), a fundamental property of the symbol (3) may 

 be demonstrated. We have 



(^a,)3,..(w)] =S±^. (^jo,/3,..(n)] (19), 



Okl^k ' ^ Pkt^k 



which when (m) is even, reduced itself by (17) to 



(^a,)3,..(«)] = [^I,^,..(w)] 2(±^±,.l) (20) 



t 

 = 1 .2... A; Jai/3, ..(w) 



akf^k 

 But when (w) is odd, from the equation (18), 



'«,, /3,..(M)| = j^«,(3,..(», 

 "*/3i ) I a/,, (it 



«) 2(± 1) = (21). 



"*/3j ) I at, /3j 



Since the number of negative and positive values of i^^ are equal. 



