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LXEII. Note on the Properties of the Test Operators which occur 

 in the Calculus of Invariants, their Derivatives, Analogues, and 

 Laws of Combination ; with an incidental application to the 

 development in a Maclaurinian series of any power of the Lo- 

 garithm of an augmented Variable. By Professor Sylvester, 

 F.R.S.f 



SUPPOSE 2 denotes any algebraical function of the two 

 sets of elements, 



/ , d d d \ 



\ a > h > C "->Ta'db'Jc>--} 



Let -xjr* in general signify the process of operating with ^ 

 upon all that follows J. 



Suppose (pi^4>i = 2 , where the operating elements -=-> -tt, • . . 



of course can only operate upon the operands a, b, c, . . . in the 

 secoud 0. In like manner, let 



0!* $!* 0! = (f> x * 2 = 3 , 



and in general 



(0 1 *)«-i0 1 = w . 



It will follow from this that 



0j# X * = (0 L 2 -f 2 ) *, 



0!* 0,* X * = (0j 3 + 20!0 2 + 3 ) * § ; 



t Communicated by the Author. 



X The symbol of an operator consists of two parts, the corpus or quan- 

 tity, and the asterisk or sign of operation. Thus a simple extensor opera- 

 tor has one of the extensors for its corpus ; a compound extensor operator 

 has any algebraical function of any number of extensors for its corpus. 

 The operator which represents the combined effect of two or more operators 

 following each other in any specified order may be termed their resultant ; 

 the theorems in the text amount to saying that the resultant of any num- 

 ber of simple or compound extensor operators is independent of the order 

 in which its components occur, and is equivalent to some third compound 

 extensor operator. One great problem to be solved is to determine the 

 corpus of a resultant in terms of the corpora of its two components. This 

 is done in the text for the simple case where each component corpus is a 

 simple power of one of the extensors. To attain clearness of conception, 

 th^f first condition is language, the second language, the third language — 

 Protean speech — the child and parent of thought. 



§ So more generally if (p, yjs be any two functions of a,b,c,... — , _,_,".,. 



da db dc 

 we have 



