Mr. CAYLEY, on THE THEORY OF DETERMINANTS. 85 



6 II. On the notation and properties of certain functions resolvable into a series of deter- 

 minants. 



Let the letters 



n, r^---r^ (1), 



represent a permutation of the numbers 



1, 2 ... k (2). 



Then if in the series (l), if one of the letters succeeds mediately or immediately a letter 

 representing a higher number than its own, then for each line this happens there is said to be 

 a "derangement" or "inversion" in the series (1) It is to be remarked that if any letter 

 succeed («) letters representing higher numbers, this is reckoned for the same number (*) of 

 inversions. 



Suppose next the symbol 



±. (3), 



denotes the sign + or - , according as the number of inversions in the series (1) is even 

 or odd. 



This being premised, consider the symbol 



\Ap,, (T, ..(w)| (4), 



denoting the sum of all the different terms of the form 



±r =•=.-• ^jOr,. 0-», >• ■^|t)r,> O-j^J (5). 



The letters 



n. i-a. ..»•*; s,, S2...S4: &c (6), 



denoting any permutations whatever, the same or different, of the series of numbers (2). The 

 number of terms represented by the symbol (5) is evidently 



(i.2...ky (7). 



In some cases it will be necessary to leave a certain number of the vertical rows p, a- . . 

 unpermuted. This will be represented by writing the mark (f) immediately above the rows in 

 question. So that for instance 



t t 

 f^,o,a, ..0,0, .(n)] (8). 



Pk'^k • • &ki>k 



The number of rows with the (f) being (,r), denotes the sum of the 



(I a. ..A)"-' (9), 



terms, of the form 



±,±,..^^,,, O-,,...0|, (/), Apr^, (T,,^...0j, <^j (10). 



This is obvious, that if all the rows have the mark (j-) the notation (8) denotes a single 

 product only, and if the mank (f) be placed over all the rows but (1), the notation (8) be- 

 longs to a determinant. It is obvious also that we may write 



[Ap,.a,..e(p,..{n)\ = ^±„ ±„.. Mp,(r, ..0„,0,, ..(w) (11), 



('k<^k--&k<()k i I Pk<^k--&",'P>; I 



where 2 refers to the different permutations, 



Ui,u.,...u^\ t),, ?)2...«j; &c (12), 



