Direct 
412 
2 : x ” (nat) #4 (bos). 
Method. 
ws = 
This is a new identical equation, which must hold true 
for every value of x. From this equation we may de- 
rive another, just as we found the last from that before 
it, and so on. ‘Thus a series of identical equations wil! 
Sa eee 
they hare been derived, will stand 
lt = 2" + (k—2)X, 
=n + (bs), : m 
efx = n (n—1) 2°-* + (k—2) = 
ss = n(n—1) (n—2) 2-3 + (b—2) er 
1X wnat) (m2) (rs) HT, 
&e. 
From these equations, » ‘wat vais one after an- 
other, the functions a ie & the follow- 
ing series of successive values ot X is obtained, 
dX 
X=onx 1 +(4—2) 
Nonx"-14 en (k—2) 
G—2) aX 
a mee Fo 
Xone + nd (k—2) 
sear aie 
&e. 
ee ae expressions for X in 
, ke=x" 4X (k—2), 
we find < 
Bampnzt) (ka) +(h—2)* 5, 
as4¢n2) (ka) + pet (hms)? 
(—zys aX 
+O 
&e. 
And again, ing r+-h instead of k, and h for 
k—w, but retai til the hypothesis that the fusion 
> dX z 
al functions >—, >—;» &c. are functions of x and &, 
ictal 2 faccmmenanteie eee 
n(n—1) 
(2+hyma"pnz— hy Sdere See ee 
And in general, supposing the series to be continued 
to m terms, (without reckoning the term that contains 
the fluxional expression ), 
(24h tan he nO) ao js 
1) (n—2 ‘ 
$ MOTO) oot 1; Be 
he aX 
+ 53....tm—l)* de" 
FLUXIONS. 
If we suppose this series to be continued indefinitely, 
then we have the binomial theorem in its common furm 
ocenan, wei meie Sele Pee, Se quan- 
Cal PAX 
23...m—1 dey 
on ta ee 
the first m terms. For example, 4* ——— expresses the 
spate, Sh all the terms following the second; and 
D * Gar °kpresses the amount of all after the third, 
and so on. ; 
When n is a whole number the series terminates, be- 
cause then ali the termsafter the canis den jenwianyrae 
In this case the expression Sn oo” be Soe ht to be 
dat gan OB 0 
=0. To veri 
recollect, that 
ai a now, when n is a whole number, the nu- 
merator is exactly divisible by the denominator, so that 
XS) pnt ems at. htt ams 
hence, considering & as constant, we have 
aX Skem-*8 42k 3243 htt... 
dx ~ L4(n—2)ka-9 4-(n—1)2-2 5 
aX f2h-342.3h—19.., 
viper +(n—2)(n—3).2*-4 4 (n—1)(n—2)2"5, 
By proceeding in this manner, we may express alPthe 
functions 4%, 2%, &e. in terms of k and 2; and it 
is manifest, that the series which expresses each, will 
have one term fewer than the series which expresses 
that before it, because of the constant quantities /"—?, 
ke—2, &c, which have:their flaxions =0. As the series 
which expresses X has terms, that which expresses 
<x will consist of n—1 terms, and that which expres- 
ses will consist of n—@ terms, and so on tothe va- 
1 ¢ OOX 
ae Fa Me 
{ (—1)(n=2)(n—-8).-t0(n—a) factors} atti Gm) 
=1.259.4... to(@—1). : 
As tle fo a cunatangapeensy ote Sealey will be =0, 
ill be = it 
thoralcee 2 ik bs sith as we Tal Gone? Gt 
er ‘ , a: 
When x is a fraction or negative, then X= 7—— 38 
still a fraction, when. the numerator and denominator 
are each divided by their common factor. In this case, 
whatever be the number m, the expression > can 
never be = 0: still, however, it may be calculated. 
Bea 1 
For example, if n=}, then X= = : 
7 hme Bt 
de ~ eet at ay de gah (Bp ady 
From the first of these expressions, we find ' 
4) 
which will consist of'a single term, viz. 
therefore, 
)% 
i 
OO EE 
