AS COMMONLY INVESTIGATED. 249 



tions as are here mentioned, problems otherwise distinct; but it is equally its 

 office to assign what groups of problems are to be separated — and to give to 

 everj form that exact limit of permanence, which will ensure a union of gene- 

 rality with despatch and clearness. 



It would add greatly to the precision of mathematical science, if the received 

 arrangements that compose the branch commonly designated as the calculus, 

 and which I have called the received arrangements of numerical logic, were 

 kept perfectly distinct from those which are yet subjects of inquiry. The lat- 

 ter are proper matters of investigation for the pioneers of the pure science, but 

 their theory should form a distinct department of analysis, to which no appeal 

 should be made in the reduction of modal relations. And in this class of doubt- 

 ful and incomplete generalizations, which it may hereafter be advisable to re- 

 ject or admit, it appears to me that we must place fractional differentials, en- 

 cumbered as they are with an infinite series of arbitrary corrections. 



A single additional remark, arising from the development we have been 

 contemplating, may perhaps be permitted me before I close this paper. I al- 

 lude to the error which several recent and distinguished writers have made, in 

 regard to the nature of the remainder involved in such expansions. The 

 formulfe very commonly used to express such remainders, give nearly the sum 

 of the developable portion of the function, reckoned from the n* term to infinity, 

 whilst it is evident that to answer the purpose for which these formulae are 

 employed, they must include, not this portion alone, but more especially that 

 which will not develope in the required form. An instance will render this 

 remark sufficiently clear. 



Assuming h-e ^^"^ for h in Taylor's Theorem, and h^e -*v/— i for h in 



\ \^ 



the expansion of ^ we have evidently 



1 r h^e -"'^~ 



-^J ■ ?, . ■_ . F(x-f.h^e^v/— )dO 

 2rt 1 — h'e "vZ-i ^ ^ 



for the value of the remainder from the n"^ term to infinity, but it is equally 

 evident, that such an expression does not comprehend the negative or fractional 

 exponents, or in short, any of those which cause Taylor's Theorem to fail; yet 

 numerous expressions of this kind are given without this limitation, and among 



VII. — 3 N 



