192 Mr. Harvey on the Theory of Arbogast. [Sept. 



A similar result may also be obtained from the last of the 

 equations marked (1). For since the arbitrary equation 

 A" = { (1 + A) - 1 }" 



maybe assumed, and that the latter member when developed 

 becomes 



(1 + A)" - « (1 + A)"-' + '^^j^^ (1 + A)»-> + 1 ; 



it follows, that by uniting the function u to the several symbols 

 of operation, there will arise 



A" u = (1 + A)" n - « (1 + A)"-' » + ~^ (1 + A)—' 



11 + u. 



But since from the equations (1) 



(1 + A)" u = «„ 

 (1 + A)"-' n = «„_, 

 (1 + Ay-'u = M„_, 



n = « 

 there will arise by substitution, 



An , « (n - 1) 



A" u = 11,^ — n 7(„_, + — j-y- /?,._, ±u, 



which is identical with the result obtained by Arbogast. 



The conciseness of the above investigations demonstrates that 

 many important advantages may be derived from the theory of 

 Arbogast. And although, in the first point of view, it may seem 

 impossible to pursue with certainty and precision any extended 

 train of analytical inquiry, by employing symbols of operation 

 when detached from the functions with which they were origi- 

 nally connected ; yet it will be perceived, that it is of no conse- 

 quence in what light we regard the symbol; whether as an 

 instrument to generate successive changes in a function, or to 

 transfer to it the momentary idea of quantity, when in its 

 detached state from the function. In the latter point of view, it 

 necessarily becomes subject to all the forms and laws of which 

 quantities are susceptible. But it may be proper to caution the 

 inquirer against his attaching a permanent idea of quantity to 

 the symbol of operation, even in its separated state. It is only 

 indeed in this latter condition that it is supposed to be subject 

 to the rules and operations of quantity, and never decidedly loses 

 the character it was originally destined to maintain in any stage 

 of an investigation. In my next communication I shall endea- 

 vour to point out some further applications of the principle. 



I am, Sir, your humble servant, 



George Harvey. 



