4416 
sed to find the area CEQP=s, (Fig. 14.) com 
Method. od between the ordinates CE, PQ, ott) de abatia- 
Fig. 14. 
“constant arbilra : quantity. 
sx are AE=a, and AQ=4 ; because d sy d x (art. 71.) 
in general, s=fy d =P + c. But let us suppose, that 
when x becomes a, P becomes A, then, corresponding 
to the particular case of s=a, s=A+c. Now in the 
present case, when x=a, then s=0; therefore, ¢ is re- 
stricted to the i that makes A+c=0, that is, 
e=—A, hence s=P—A ; and if in this expression we 
+ instead of x, we have the area, or the value of s, 
wom the Henke of ene ae oo ; : 
in, us su u x are two variable 
qiatities, ov telaten, that der ds, then, in general, 
ate But let us farther suppose it known, 
c= 
. a"+1 
that when r=a, then u=/. In this case, k= aqit® 
and hence c—k— at and, in the question under con- 
sideration, it can have no other value; therefore « = 
pil on +k. In this manner, we may determine the 
n+ 
value of the constant correction c. 
Supposing a to be the value of zx, when the fluent 
=0, then a is called the origin of the fluent ; and it is 
said fo begin when x has that value ; and to end when 
x has completed the change in its magnitude, so as to 
have from r=a to x=b. These values of x are 
the /imits of the fluent. When neither the origin nor 
joe limits of the flu- 
ent are , It is definite. us, supposing A to 
be the value of the i ge 2=.4, on B its value 
when r—6, then B—A is the definite fluent. As c has 
the same value in A and B, it disappears from the de- 
finite fluent, which may therefore be found by putting 
2=a, and x-=b in the indefinite fluent, and subtracting 
the first result from the second. All this will be illus- 
trated by examples as we proceed. 
Fluents found by Infinite Series. 
148. When a fluent cannot be assigned in finite terms 
by algebraic quantities, nor by circular arcs nor loga- 
rithms, then recourse must be had to methods of ap- 
proximation. Let fd be the fluent. If we develope 
the function < into a series, proceeding according to the 
ascending or descending powers of x, and multiply the 
terms by d x, and then take the fluents, their sum will 
evidently be the fluent sought. We shall now give ex- 
amples. 
Ex. 1. Let the fluent of A be required, which, 
we know, comprehends in it ].(a+.2), (Rule(B), art. 
26). acbaale derision, fot ere 
1 we igts, 2 
Saat al sami valet Oe. 
Therefore, multiplyi dx, 
> otorereacadeg jon, Aad 
dz z F id 
a agar Seed 
f sea a ear t sa Tere Pa 
In the most general expressiun for the fluent, ¢ ma 
be any constant quantity whatever; but regarding it 
FLUXIONS. 
as expressing the value of |. (2+4-a), ¢ is restricted to a 
particular value. pee acy ig arama 
when 1=0, then L (+e) becomes 1. (a); but when 
x—0, all the terms of series except c vanish, there- 
fore c= 1, (a), and 
2 # x 
1, (v4) =L.@+o—eat 5qs — &e- 
as we have already found, (Art. 53.) é 
Ex. 2, To find the fluent of 5", we expand it by 
division into the series dt — 2*dx4atdzr—2x°dzx+ 
&c. then taking the fluents of the terms, we find 
dx , 
sr—tz pig 
i4e ile i) ch i ath db 
As this fluxion is that of an are, of which the tangent 
is x, and radius unity, (Art. 35), by giving a suitable 
value to c, the fluent must express that are. To 
determine c, we must consider, that when the tangent 
= 0, then the are =0; therefore the fluent ought to va« 
nish when x0; hence c must be =0, and 
arc (tn. =2) = 25 +S 48. 
which is the series origi found by James Gregory, 
and perhaps prec 25 Leibnits,” Hence the ratio 
of the diameter to the circumference may be found, wee 
Anitumetic of Sines, Art. 82); and the 
$.14159265 ... which expresses the measure of two 
right angles, and is commonly indicated by the cha~ 
racter x, ’ 
d 
Ex. 3, Let the fluent of F ; = jorda(1—et) 
be required. In this case, we must the radical 
into a series by the binomial theorem, (Art. 53. or 54. 
See also Atcesra, Art, 323.) which will give 
a 1.3.24 
ary f=14¢ + gq + &c hence we have 
dx x3 S x6 3,5 x7 
Vas) =*+ 93 + ong t or67 t St 
This fluxion is the same as that of an arc, of which 
the sine is x, (Rule (D) Art. 26.) ; now the are vanish~ 
ae Antone by i Ata eee ie! 
ill express the arc. If we suppose arc to be ¥ 
the circumference, or 30°, then «= }. "Therefore ob« 
serving, that the are of 30° is ¥ #, we have 
gical 1 1.3 1.3.5 
t7=3+ 939 + case t+aae70 + & 
149° John Bernoulli invented a eral expression 
for any fluent, which is 8 Taylor's formula 
for any function of a binomial. Let z be any function 
of x, then any fluent whatever, containing only one 
variable quantity, may be represented by f/zda: em« 
ploying now the formula fu dt = ut—ftdu, we have 
prdzaz2—fadz 
fotenfierimiot fae 
a Fei! a as 
and taking the fluent ye eee ey ee 
az dD z 
ad 
Tat des jt atl at 
tnstead of f'rd, ft FE, Se. substitute thei 
dis 
dz® 
&e, 
