412 Prof. Cayley on a Property of Commutants. 



or, in a notation analogous to that of a commutant, 



+A+' 

 ^i 1 



2 2 

 L_ p p-i 



denote a function formed precisely in the manner of a determi- 

 nant (or commutant of two columns), except that the several 

 terms (instead of being taken with a sign + or — as above) are 

 taken with the sign + : thus 



{^11 X 12 ~") 

 X 2 1 X 22 J 



or L Si] 



each denote 



A<ilA 22 + A, 12 \ 2 1« 



This being so, the theorem is that the commutant 



A 1 



"""I 1 l..(0) 

 2 2 2 



where 



>'S t .. (0) 



p pp 

 = rX lr3 \ l8 . . (0) ^ 



A>2n ^2s 



\) rf \j s 



rlt 



*3 I 



P J 



whenever p> 0, is =0. 



To prove this, consider the general term of the commutant, 

 viz. this is 



±,±f-K,-r 



2 s t . . p Sptp . , 



the general term of A r s t . . is \ a r \ s X c t . . , where a, b, c . . repre- 

 sent some permutation of the numbers 1, 2, 3 . . 6. Substituting 

 the like values for each of the factors A l8 , t , , A 2s „ r , &c, the 

 general term of the commutant is 



= ±s± t 



. A, , , A, , „, A. . ., 

 a 1 bsc t 



\"^\"s"\"t" ' ' ^a p p\ p s p \ P t p ' 



Taking the sum of this term with respect to the quantities 

 s', s", . . s pi which denote any possible permutation of the numbers 

 1, 2 . . . p ; again, with respect to the quantities /', t n . . t p , which 

 denote any possible permutation of the numbers 1, 2, . . .p; and 

 the like for each of the (0—1) series of quantities, the sum in 

 question is 



\'l\"2- 



\ p p%± S \<S> \>8<<- 



\s p ^±t\'t'\"t X c p t p "> 



