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VIII. On the Multiplication of Partial Differential Operators. 

 By Professor Sylvester*. 



IN the last Number of the Magazine I explained the sense in 

 which I employ the term operator as distinguished from an 

 operant, the distinction being somewhat analogous to the gram- 

 matical one between a verb and a noun ; for as a combination 

 of the predicate and copula gives rise to a verb which has 

 independent laws of inflexion and regimen, so an operator is 

 a new species of quantity which, springing from the union of 

 an operant and the symbol of operation, becomes amenable to 

 its own proper laws of functional action and subjection f. I found 

 it convenient also to refer to an operator as an energized operant%. 

 At the outset of the paper a proposition was stated inadvertently, 

 regarding any energized function of a set of variables and their 

 corresponding elementary operators, in too general terms. Such 

 function remaining unrestricted in regard to the principal letters 

 Xj y, z, . . . should have been limited to be a linear quantic in re- 

 gard to the elementary operators 8 X , h y , 8 Z) . . . If </> be any such 

 function, the proposition in question, thanks to the happy intro- 

 duction of the star symbol, may without any auxiliary definition 

 of the derivatives <£ 2 , </> 3 , . . . employed in the preceding paper, be 

 stated as follows, with perfect freedom from any shade of am- 

 biguity, 



which theorem (t being an arbitrary parameter) contains the 

 general rule for expanding (<j)*) n in terms of the quantities 

 [<£*$]#; [</>*$*</>]*; [<£*<£*<£*$]*, &c.§ 



* Communicated by the Author. 



t Thus an operator forms a new part of speech in algebra. It may be 

 well to notice in this place, in order to prevent error arising hereafter, that 

 the process of energization must in general be indicated, not by the mere 

 apposition of an asterisk, but of brackets and asterisk. Thus, although 

 P turned into an operator may be correctly designated by P*, P*P similarly 

 energized will be represented by [P#P]#, and riot by P*P*. 



Conversely, denergization will consist in the abstraction of an asterisk 

 and brackets, and not of the former merely. Thus P*P* denergized 

 is not P*P but P 2 +P*P, because P*P* is [P 2 + P*P>; whereas P*P* di- 

 vided by *, a term employed in the sequel in a footnote, is simply P*P, so 

 that star division, or destellation as it may be termed, is not to be con- 

 founded with denergization. 



\ Or I might have used the word vitalized to convey the same idea, — the 

 operator being the operant endued with power of action, but none the less 

 for that capable of being acted upon, calling to mind the relation between 

 dead and living matter. So denergization might be termed amortization, a 

 word which exists in the language. 



§ Thus, ex. gr., when 



(f)=bd a + 2c8i,+3ddc+ . . . , 



