96 Application of the Fluxional Ratio, &e. 
order; when we wish to express it sicorallt ja x may be written [a]?, 
and x may be written [z]* and so on. Suppose that E represents 
any power x” generally, the fluxions of the several orders are express- 
ed in the following manner. In the preceding case stands for the 
exponent 4, and E stands for x* ; [E]? for 4°, the first fluxion ; 
[E]? for 12%?a°? the second fluxion; [E]* for 24xz-? the third 
fluxion; [E]* for _ * the fourth fluxion; hence 4x°x*++ 
12n72°?  24xax"* — E E}? [E 
eae Aa, aS 9) IY (EI, [EI 
ment. The larger the ene n is taken, the awe il be the 
number of terms, of which the series is composed. When n is in- 
definitely large, the series becomes infinite, and in that case E stands 
for a"; [E]! for na*“'a°; [E]? for n(n—1)a*-*2°2 ; [E]* om: 
n(n—1)(n—2)a"-%2°3; [E]* for n(n—1)(n—2)(n—3)x"-4a"* 
[E]* for n(n—1)(n—2)(n—3)(n—A) a= #28 "5; &c. The sities 
expressing the orders of fluxions becomes (1. )fE}! +[E}? +[E]° + 
[E]*+[E]*+[E]°+&c.... in inf. and the series expressing the 
increment becomes : 
[E a pike [E]*  [E]* 
(2.) (e-bay = [E)' + oF og te54t254.5 te 
in inf. -=increment. 
The series of Maclaurin is 
E ey el sy ee 
(3) y= E+ tt ges 14 a3e8? Fasars* a 
a el | 
2.3.4.52°° 
The series of Taylor i is, 
a Dk h we h Aoi [yl 
(4) fle tia ee Se te aga the. 
The binomial series is, 
=incre- 
(5.)(a+h)"=2" +nx"-"h--n. ethan. om Set et 7 hap ee. 
If in the series of Taylor we make i—[E], i x*=1,and in 
the series of Maclaurin, if we make z and z- each, equal to 1, they 
will coincide with the preceding series. These series indicate, what 
share each of the orders of fluxions has in forming the increment, 
and disclose the relation of the several orders of fluxions to the flu- 
ent, included in the following properties. 1. When z+-2’ represents 
