UEPOKT ON CERTAIN BRANCHES OF ANALYSIS. 223 



The law of derivation of the terms in Taylor's series, 



" = ^ + ^ • ^'' + ^- O + ^^ • 17273 + ^'■' 



is the same as in the more general series 



dx" ~ dx' ^ dx'-+' ' 1 "^ dx'^^' 1.^2 ^*' 



and if we possess the law of derivation of -7 — and of -1—7, we 



can find all the terms of both these series, whatever be the 

 value of r. The first of these terms must be determined through 

 the ordinary definitions of the differential calculus ; the second 

 must be determined in form by the same principles, and gene- 

 ralized through the medium of the principle of equivalent 

 forms. Both these processes are indispensably necessary for 



d^ u . . 



the determination of -r-z '• but it is the second of them which 

 dx*^ 



altogether separates the interpretation of -T~r from that of -7—, 



or rather of ^-— . w^hen r is a whole number, unless in the par- 

 es j;*^ 



ticular cases in which the symbols in both are identical in 



value. 



There are two distinct processes in algebra, the direct and 

 the inverse, presenting generally very different degrees of dif- 

 ficulty. In the first case, we proceed from defined operations, 

 and by various processes of demonstrative reasoning we arrive 

 at results which are general in form though particular in value, 

 and which are subsequently generalized in value likewise : in 

 the second, we commence from the general result, and we are 

 either required to discover from its form and composition some 

 equivalent result, or, if defined operations have produced it, to 

 discover the primitive quantity from which those operations 

 "have commenced. Of all these processes we have already given 

 examples, and nearly the whole business of analysis will consist 

 in their discussion and developement, under the infinitely varied 

 forms in which they will present themselves. 



The disappearance of undulating and of determinate func- 

 tions with arbitrary constants, upon the introduction of inte- 

 gral or other specific values of certain symbols involved, is one 

 of the chief sources * of error in effecting transitions to equiva- 



* The theory of discontinuous functions and of the signs of discontinuity 

 will show many others. 



