59 



result will be that obtained by differentiating with d y and multi- 

 plying by x. And if the operation upon the elements tantamount 

 to xd is represented by {xd ' }, then writing down the series of ope- 

 rations 



x'd y i, . .&c., 





where x, y are considered as being successively replaced by every 

 permutation of two different facients of the set (a?, y..), x',y' by 

 every permutation of two different facients of the set (x',y' ..) &c., 

 then it is clear that the quantic is reduced to zero by each of the ope- 

 rations of the entire system, but this property is not by any means 

 confined to the quantic ; and any function having the property in 

 question, i. e. every function which is reduced to zero by each ope- 

 ration of the entire system, is said to be a covariant of the quantic. 

 The definition is afterwards still further generalized, and its connec- 

 tion explained with the methods given, in the memoir ' On Linear 

 Transformations,' Camb. and Dub. Math. Journal, Old Series, t. iv., 

 and New Series, t. i., and the ' Memoire sur les Hyperdeterminants,' 

 Crelle, t. xxx., and some other theorems given in relation to the 

 subject. The latter part of the memoir relates to the theory of 

 the quantic (#)(#, y) m , and to the number of and relations between 

 the covariants, and as part of such theory to the beautiful law of 

 reciprocity of MM. Sylvester and Hermite. 



