Application of the Fluxtonal Ratio, &ec. — 101 
In the equation (3) p is considered as a function of z. Let p be 
represented by f’x, and p’ by f’z, &c. Inasmuch as the develop- 
ments (2) (3) are identical, the terms which contain the same pow- 
ers of A are necessarily equal; consequently if we compare the 
terms of these two series affected with th, ih?,ih*, &c. we shall find that: - 
2q=p’=f"x, hence g=}p/=1f"r. Again, 3r=q’=1f’’z, hence 
r=37 =74f 2. Again, 4s=r =}.5f"2, hence s=jr =}.4.4 fa. 
Again; Ste ecty.ff se, hence teeje=}.1..0f" 2. “a thus 
proceeding we shall find successively all the other coefficients of the 
equation (1). Substituting in this equation the values of p, q, r, s, &c. 
. 4 
: h2 ‘ 
we shall have f(2--h)=frtfichtf'rg t+f"'r 93 th"t9 3.4 t 
hs. | 
f "53.4.5 +&c. (4.) This series is of a very general nature, com- 
prehending that of a power of a common binomial quantity indicated 
by («-+-h)" ; that of a power of a logarithmic binomial quantity indi- 
cated by log. (x-+-A)”, and that of a power of a binomial sine or co- 
sine indicated by sin.(z+-h)", or cos.(z+-h)". In the equation (4.) 
J’, being 'the coefficient of h in the — of the common bi- 
nomial quantity, is indicated by a2"-'; fz, being the Sane of 
h? 
9? is indicated by Bats; ; flr, bate the coefficient of-—q Is in- 
dicated sd yx"~*, and so on: hence putting these values of ae f LE 
fz, &c. in the — (4), we have f(z+h)=2"+a2"""h+ 
2 a 
Rar? yn"? 23 +z" nas 3.4 + &e. (5.) which is the common 
binomial series. 
The several orders of fluxions are produced by multiplying their . 
fluxional ratios, each, into the preceding order, that is to say, 
a" X—> =n2""12, the first fluxion. se tlilias 
se a 
pe" ee =n(n—1)2"- *x*?, the second fluxion. (7.) 
A : 
n(n — 1)r*- 3 et aces a = n(n—1)i(n—2)x"-%2-%, the third 
fluxion. (8.) 
_ Now if we consider equation (b. ) we shall perceive that the pr 
ficient of h, in the second term in the development of the binomial 
(2+h)",is ax"-' ,and is the same with the fluxional coefficient nz" ~' 
