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LVITI. On a Property of Commutants. 

 By Professor Cayley, F.R.S.* 



CALL to mind the definition of a commutant, viz., if in the 

 symbol 



1 1 1 (9) 



2 2 2 



P PP 



we permute independently in every possible manner the numbers 

 1, 2, . . .p of each of the 6 columns except the column marked 

 (f), giving to each permutation its proper sign, + or — , accord- 

 ing as the number of inversions is even or odd, thus 



+ .+ 



which is to be read as meaning 



2 s 2 t 2 

 ps p i p 



± s ± t ---K 



.A. 



Sp tp. 



2 s 2 t 2 . . 



the sum of all the (1.2.3.. .p) e ~ l terms so obtained is the 

 commutant denoted by the above-mentioned symbol. In the 

 particular case 6=2, the commutant is of course a determinant : 

 in this case, and generally if 6 be even, it is immaterial which 

 of the columns is left unpermuted, so that the (t) instead of 

 being placed over any column may be placed on the left hand of 

 the A; but when 6 is odd, the function has different values 

 according as one or another column is left unpermuted, and the 

 position of the (f) is therefore material. It may be added that 

 if all the columns are permuted, then, if 6 be even, the sum is 

 1.2...JP into the commutant obtained by leaving any one 

 column unpermuted; but if 6 is odd, then the sum is =0. 



The property in question is a generalization of a property of 

 determinants, viz. we have 



2XV , XfJ + TJfi, 



Xjd + \ ! /jl, 2/jl/J , 



Xv 7 H-XV, fiv 1 ' + fjJv , 



\v ! + X'v, . 



fjbV 1 +fjJv, . 



2vv' , 



= 



whenever the order of the determinant is greater than 2. 

 To enunciate the coresponding property of commutants, let 



(^ut X 12 . . ""I 

 X 21 , X 2 2 f 



* Communicated by the Author. 

 2E2 



