AS COMMONLY INVESTIGATED. 235 



When this is borne in mind, it will be readily understood that if the secon- 

 dary terms are not of the same magnitude with any of the terms in the series ; 

 in other words, if we have classed all the secondary terms of the different func- 

 tions that may happen to agree with the primary term of X or Z, as a part of 



n II 



the latter, it will follow that our equation (7) cannot subsist unless we have 



separately 



5 P X = Q Z ? 



( n n n n ) 



X = a 



for every value of n. 



The restriction here supposed, however, will be unnecessary when the equa- 

 tion is of the form 



PX + PX + &C. = 



11 2 2 



since the result P = 0, already demonstrated, will also lead to P = 0, and 



thence, generally to 



P = 



n 



which is true for every value of n. 



The two great forms of arranging terms which algebra employs are drawn, 

 in common with nearly all its arrangements, from arithmetic, and are simply 

 series connected by the operations of addition, or of multiplication; whence, 



12 



denoting by X, S, S, prefixed respectively to the general term of a series U, the 

 three arrangements 



u, u, u, u, 



12 3 



U + U + U + U + &C. . 

 UxUxUxUx&c. . 



12 3 



we shall have a concise notation for all the principal arrangements or develop- 

 ments of which it will be necessary to speak. 



The operations of every branch of analysis consist, in a great degree, of find- 

 ing equivalencies between the two last of these arrangements, and of thus, 

 either by the species of correction employed between the terms, or by the na- 

 ture of the classifying function U, suiting our analysis to the problem in hand. 



X 



Taylor's Theorem is one of the transformations of this kind, and perfectly 

 to comprehend it, and especially to connect it with the elementary operations 

 of the science — a task quite as important, we must consider not only such terms 

 as enter into the series (2,) but also those which compose series of the class (3;) 



u. . 



. &c. 



(1) 



u.. 



. &c. 



(2) 



u. . 



. &c. 



(3) 



