248 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



of h, multiplying by k^'-"', differentiating n times, and making k equal to zero. 

 Whence 



P = ^ ^^ ^. d^. k"'^' F (x + k'^) (12) 



We have already remarked that the expansion of Fz by powers of z — x seems 

 to be more truly the algebraic development, than the form vv^hich this expan- 

 sion assumes when presented as Taylor's Theorem — the one is the develop- 

 ment of a general monomial, the other a general trinomial function — and it 

 seems a more natural order to deduce the latter from the former, than to fol- 

 low a converse process. The substitution of z — x for h does affect the coeffi- 

 cients, and thus we may write 



F.z = S (^ — ^> . d:\ 5kP>'. F (x + k^)5. (13) 



1.2 n KC ^ /J„ "- ' 



k ^ 



where p' is minus, the numerator of the least value of n, supposed always to be 

 zero unless n is negative — and [i is the least number divisible by the denomi-- 

 nators of n. 



I shall not increase the size of this paper, which has already extended itself 

 to such undue limits, by exemplifying the use of this theorem, in determining 

 the values of vanishing fractions, or in general of functions at their critical values; 

 but I may remark here the light which it throws on fractional differentials. 



The proposition 



F (x + h) = S d^ F (x). ^" 



1.2, 



is certainly that from which we obtain our only notion of a differential coeffi- 

 cient; and thus, were we to embrace Mr. Peacock's theory of the permanence 

 of equivalent forms, it would follow that d" F x was the coefficient of the n* 

 term in the expansion of F (x -f- h) whether n was positive or negative, whole 

 or fractional; and, consequently, that in all but a certain class of transcendents 

 d^ F X was zero whenever n was other than whole and positive. Such a re- 

 sult would be at variance with the received theory of fractional differentials, as 

 would likewise happen with the values given by that theory and the formula 

 (12) in those cases where the fractional powers appeared in the expansion. 



I do not mean to deny that artifices and conventions can be made, and arti- 

 fices and conventions that are perfectly allowable, whereby these seeming dis- 

 crepancies would be reconciled ; but I hold that such a process belongs to the 

 doctrine of correlations — a doctrine greatly misunderstood by the writer in ques- 

 tion. It is the office of correlations to group together by means of such conven- 



