244 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



S (x — r) = S . P x^ * 



z = c|n z z = oln 



wherein all the factors are equal. Such an equivalence we know is just for 

 every value of the variable, whereas when, by analogy, or any other species 

 of proof, we extend the like proposition to negative or fractional indices, the re- 

 mainder must now be taken into account — or otherwise the proposition must 

 be limited to values of x that make the right hand member converge, when 

 one equation becomes 



5 S (x — r) = 2 P x^ ? 



fz = l|n n z = 0|n S 



X ^ — X . . . + I 



The same error, of course, occurs wherever remainders are neglected; neither 

 is the error here exactly of the kind that I have been noticing in Taylor's 

 Theorem; but I have instanced it as a very familiar case in which, also, our 

 usual demonstrations prove too much. 



That whatever functions are used in Taylor's Theorem in place of the 



h" 

 classifying function j-g — -, they must be of the same order with the latter, 



will be seen by considering that the form arrived at was deduced by such 



mere transpositions of secondary parts as did not alter the arrangement of the 



several terms. 



And having thus shown that this celebrated development is truly the de- 

 velopment of monoramic functions of the first order of continuity, and accord- 

 ing to the prime factors of their several parts, I shall now proceed, in a few 

 words, to consider how far the expansion is prime. This condition is essential 

 to all the applications of the theorem, and yet no other notice of it is taken in 

 our text books than such as relates to the terms obtained by integration, or 

 those on the left hand of a given term, when the development proceeds from 

 the latter, and with regard to which an omission is always made. 



That the series derived from the left hand to the right is prime, immediately 

 follows by the usual reasoning employed to show that series are identical. For, 



z z 



if F (z -f h) would expand in two series X P h% and 2 Qh'^, we should have 

 the equation 



S (P — Q) h^ = 



2 1 



* 2 expresses a series of factors, as i: or 2 does of terinsi 



