Application of the Fluxional Ratio, &c. 99 
: ed n nn—l n=2 
the third and fourth terms, we have re * , and =° at Sade dest 
which are the unciz of the same terms in the cat 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 may further serve as an apology for saueiree it that, which 
has long been’ known. 
Let f (c-+h) represent generally a function which has not yet 
been reduced to a series. To convert this function into a series we 
may suppose, , 
“fle+h)=fet Ph 
P= p+Qh 
Q=q+Rh 
R=r-+Sh 
Substituting for P, Q, R, S$, &c. their several values, we have, 
Sath)=fet+ph+gh? +rh3+sh* +th®+&c. (1.) 
In any binomial A (z+h), if 2 is changed to x-++1, it will give the 
same result, when raised to a given power, as it will, when / is chan- 
ged toh+i. For since the root A(z+i+h) is the same with the 
root A (x-++-h+7), they will yield identical results, when raised to any 
proposed power.* Hence it follows, that in the development fz+ 
ph+-qh? +rh? +-sh* + &c. we may first change A into h+1, and after- 
wards z into z+4, and still the two results will have the same bins 
Substituting 4+7 for h. 
The series marked (1.) in this case becomes, 
fet (h+i)=fetp(h+i)tg(h+i)? +r(h+i)e+ s(h + i) + &e. 
and writing only the two first terms in each of these binomials we 
have fle (i+) =fe + ph-+pit-gh? +-2ghi+-rh’ +3rh7i4+-&e. + ) 
Substituting 2+-+2 for x. 
