303 Mr. J. J. Sylvester on the Relation between the 



I have in previous papers detined a " Matrix " as a rectangular 

 array of terms, out of which different systems of determinants 

 may be engendered, as from the womb of a common parent ; 

 these cognate deten\iinants being by no means isolated in their 

 relations to one another, but subject to certain simple laws of 

 mutual dependence and simultaneous deperition. The condensed 

 representation of any such Matrix, according to my improved 

 Vaudemiondian notation, will be 





To return to the theorems of the text. Theorem (2.) admits 

 of being presented in a more convenient form for the purposes 

 of analytical operation, so as to become relieved from all cases 

 of exception appertaining to particular terms. 



The limitation to the generality of the expression for Q arises 

 from oui' treating 



as identical with its equal, 



Ua 



an J 



"6, "e^ 



If, however, we now convene to treat these two forms as distinct, 

 so that in theorem (2.) 



■s — ^ — 2 ^ — ^ f terms, then we may 



will contain 

 write simply 



«/>. <^2 . . . C^,. I be^ be^... bg^, f 1 b^^ b^,^ . . . b^/' 



which equation is subject to no exception for the case of the ^'s 

 and <^'s becoming identical. As regards this theorem, it will 



substitutions — a problem which appears to set at defiance all the processes 

 and artifices of common algebra. I have succeeded m applying a method 

 founded upon this calculus to the linear reduction of a biquaihatic function 

 of two letters to Cayley's form x*-\-mxh/+y'^, and of a 5" function of two 

 letters to the new form x''+y^ + {ax + hyf. This last reduction is eflTected 

 by means of the jjroperties of a certain other function of the 8th degree 

 connected with the given function of the 6th degree. See a paper on this 

 subject in the forthcoming May Number of the Cambridge and Dubhn 

 Mathematical Journal. 



