Application of the Fluxional Ratio, ^c. 99 



,,.,,-, , n n — \ , n n—l n-2 



the third and fourth terms, we have y- —^, and y — n~' — o~> 



which are the uncias of the same terms in the binomial series, and 

 the two series coincide. These properties apply equally to the re- 

 maining terms. Hence, by the aid of the binomial series, the nature 

 of the several orders of fluxions is indicated. 



Lagrange, after considering the great utility of the theorem of 

 Taylor in explaining the nature of fluxions, succeeded in demonstra- 

 ting it without making use of the Calculus. Thinking it may be 

 acceptable to those readers of the Journal of Science, who have a 

 taste for the mathematics, but have made no great proficiency in pur- 

 suits of this kind, to see a demonstration of what I believe to be the 

 true foundation of fluxions, brought down to their capacities ; I have 

 extracted from Boucharlat the method of deriving Taylor's theorem, 

 invented by Lagrange. The process is simplified, and an ellipsis is 

 supplied, necessary to an" easy understanding of the demonstration ; 

 which rnay further serve as an apology for introducing that, which 

 has long been known. 



Let f{x-\-K) represent generally a function which has not yet 

 been reduced to a series. To convert this function into a series we 

 may suppose, 



P=p-f-QA 



R=r-|-SA 



Substituting for P, Q, R, S, &;c. their several values, we have, 

 f{^j^h)=^fx-^ph+qh-'-\-rh^-\-sh'-^th'+Uc. (1.) 



In any binomial A {x-\-h), if % is changed to x-{-i, it will give the 

 same result, when raised to a given power, as it will, when h is chan- 

 ged to h-{-i. For since the root K{x-\-i-\-h) is the same with the 

 root A {x-\-h-\-i), they will yield identical resuhs, when raised to any 

 proposed power. Hence it follows, that in the development fx-\- 

 ph-\~qh^ -\-rh^ -{-sh* -{-he. we may first change h into h-\-i, and after- 

 wards X into x-\-i, and still the two results will have the same value. 

 Substituting A+i for h. 



The series marked (L) in this case becomes, 



f{x-\-{h-\-i)=fx-\-p{h-{-i)+q{h-{-iy' -\-r{h-\-i)''-}- s{h + i)' + &c. 



and writing only the two first terms in each of these binomials we 



h&vef{x-{-{h-\-i))=fx+ph-\-pi-{-qh^+2qhi+rh^-{-3rhH-{-hc.{2.) 



Substituting x-\~i for x. 



