466 Prof. Sylvester on the Properties of the Test Operators 



name for the former. Accordingly, in conformity with the 

 general terminology of the new algebra, I propose to substitute 

 the name of Protractor for Extensor to signify the operator, so as 

 to be able to use the word Protractant to signify the corpus. 

 Also I shall give the analogous names of Pertractant f and Per- 

 tractor — the former to the lineo-linear function above referred 

 to, the latter to this function energized, i. e. converted into an 

 operator by the addition of the asterisk %, the symbol of opera- 

 tive power. 



We thus start with a pertractant Pj which is energized into a 

 pertractor, P x *; with this latter we continue to operate any number 

 of times upon the original pertractant, and obtain a succession 

 of new derived pertractants, into which it appears at present to 

 be convenient, for the sake of uniformity, to introduce the 

 numerical divisors 2, 3, 4, . . ., so that we may define P»+i, the 



(P*) W P 



nth derivative pertractant, as equal to Jl. • 



We thus obtain a series of pertractants, V v P 2 , P s , . . ., which 

 may be termed the primitive and prime derivative pertractants of 

 the family. 



Again, we may form any algebraical function of the primitive 

 and its prime derivatives, and such function may be termed a 

 compound derivative of the family; this in its turn, by the 

 addition of the symbol of operative power, may be energized 

 into a pertractive operator, which, containing only a single aste- 

 risk, is to be regarded as a simple or single derived pertractor, 

 although its corpus is a compound derivative. 



The first leading proposition of the theory is, that all operators 

 so formed are commutable, so that, being subject to the laws of 

 algebraical operation, they may themselves be made the subjects 

 of algebraical functions. The second great proposition is, that 

 any such function of one or more pertractors is reducible to the 

 form of a single pertractor, i. e. is an energized function of the 

 prime pertractants P 1? P 2 , P 3 , . . . 



The theorems that have been stated concerning protractants 

 and protractors will continue to subsist for the much more 

 general class of pertractors and pertractants. Thus, ex. gr., 

 theorem (D) in the text above, when we take /a = 1, becomes 



E{* Ef* ==El +J r + i .iEl + ^ 2 (2E 2 ) + A — fH[ l Ei +y - 4 (2E 2 ) 2 + . . 



t Thus the " Universal Mixed Concomitant" x j- +y-j- +2-3- + ... is 

 of the genus Pertractant. 



