462 Prof. Sylvester on the Properties of the Test Operators 



and in the general case there will be found no great difficulty 

 in obtaining the following theorem, 



(^i*)*=nf . coefficient of t* in T, 

 where 



T=<M+<feo +< k do + "-' • * (A) 



a relation which may be expressed by means of the identity 



e«*i» = (e T )* t, ........ (A) bis 



which important equation has been previously noticed by Pro- 

 fessor Cayley under a somewhat less general form. 



With the exception of noticing that (4> l *) r and (^> 1 *) s are 

 commutable symbols by virtue of their definition, i. e. that 



(^.*)W) s = «>!*)* w>i*r, 



I am not at present aware that this theory of derivation when 

 the form of cj> is left undetermined presents much that is re- 

 markable. Very different, however, is the case when we proceed 

 to give to <p the particular form in which it enters into the cal- 

 culus of invariants : a most surprising and unexpected system 

 of relations then springs up between the various orders of ope- 

 rators ; and a vast and inexhaustible theory opens out before us, 

 of which I want leisure to be able to do more than briefly notice 

 one or two salient features. 

 Let 



•n d rt7 d, n d , d rt7 , d , a d 



and similarly 



Hence if two operators $*, ^* are commutable, so, in respect to the sym- 

 bol of operation -& , are the two operants <fi, \js. 



The force of the bracket explains itself. This wonderful symbol has the 

 faculty of extending itself without ambiguity to every possible development, 

 however new, of mathematical language. It is susceptible only of a meta- 

 physical definition as signifying the exercise, with regard to its content, of 

 that faculty of the human mind whereby a multitude is capable of being 

 regarded as an individual, or a complex as a monad. In a word, it is the 

 symbol of individuality and unification. 



t Thus, ex.gr., let 4> x represent x ~r, then (f) 2 , <p 3 , . . will be all equal 



to (j^; accordingly T=(e' ; — l)$ x , and the formula in the text becomes 



a remarkable formula of expansion. 



