FLUXIONS. 
the fluxion of the root a 4 —* 4 is — 4x 3 x ; hence the fluent 
4 S 7 
fx— 4 * 3 -v ' 32 
Ex. 4. What is the fluent of 
— J Thisquan- 
<L>-f6.x 9 |2 
tity is 4-6 a 9 |— \ yx s x ; and the fluxion of tlie root 
a* + 6x 9 is 54x 8 jrj therefore the fluent is 1~ X.x 8 x 
iX54* 8 * 
27 
Quantities which at firfl do not fland under this form, 
may frequently be reduced to it. 
dx 
Ex. 5. What is the fluent of jt—— Firfl, a 2 4 - 
a-{-x 2 [2 
yx-—u 2 x 2 + tXat 2 ; therefore a 2 +-- 2 ] f = 
a-x ^-h-iJ-X^ 3 » hence. 
y 
_ a 2 x— z + i)\Xx 3 
x + l l 2 X fix x, where the index of x without is 
lefs by unity than that under the vinculum ; hence, the 
fluent is + X 3 x __ 
—iX —2 a 2 x 3 x 
a 2 x~ 2 + i\2Xa 
a 2 yx 2 \^ya 
40. 11 both quantities under the vinculum be variable, 
and the quantity without be the fluxion of the quantity 
under the vinculum, or in a conftant ratio to it, the fluent 
may he found by this rule. Thus, the fluent of 
ay- +_y 4 jT x isix^y+^l but thde 
3 
cafes feldom occur. 
Prop. XVI —To find thcfuent of afi-cz”)’”dz r " — 'i, where 
the index of z without the vinculum increafed by unity , is fome 
multiple of the index of z under the vinculum. 
41. Put a+cz- = x, then z*= — 
c c r ’ 
take its fluxion,and rnz n ‘ — r Xx—a\ ^ ^ r „, 
—X 7 ^a) ‘x-vj hence, (putting r——s ,) dz rn —*z— 
d _ d 
— Xx—aj xx — (by expanding x=Tn — x x X 
____ J nc 
x : sax 3 —— a 2 x y 2 —, See. fubftitute this quan¬ 
tity for dz rn 1 x, and x m for a-j-cz 1”, and the given fluxion 
is transformed to — X x"’x x 
nc r 
2dly. If r be a pojitive whole number, and ?n a negative 
whole number greater in magnitude than j+i, or r, the 
fluent can always be found. But if m be a negative whole 
number equal to or lefs in magnitude than r, the denomi¬ 
nator of one of the terms mult become —o, in which cafe 
the fluent of that term fails; for in the fluxion it was of 
this form x 1 .v, which (by Art. 38) admits of no fluent 
by the rule here given; it may, how ever, be found by 
logarithms, as will be explained in Art. 45. 
3dly. The eiven fluxion, by reduction, becomes 
az —" 4 -rJ" X dz X " hence, if m and r be both 
fractions, but fuch that m-\-r may be a whole negative 
number, the fluent can always be found. This will ap¬ 
pear, by transforming the fluxion as before ; and the feries 
will always terminate ; nor can any of the denominators of 
the terms of the fluent become equal to nothing, fo as to 
make the fluent of fuch term fail. 
To find FLUENTS by LOGARITHMS. 
42. The property of logarithms, or their relation to 
natural numbers, as has been already explained in Alge¬ 
bra, is this, that as the natural numbers increafe in geo¬ 
metric progreflion, their logarithms increafe in arithmetic 
progreflion. 
43. Let a increafe till it becomes />, c, . . . . m, no, Sc c. 
and fuppofe a : b :: b : c :: &c. :: m : n :: See. then a : m :: 
a—b : m—n ; now', a—b is the increment of a, and m—n 
is the increment of m ; hence, a : m :: t he increment of a : 
the increment of m ; and as this is true in every flate of 
the increments, if we make them vanifli, we have a : tn 
as the limiting ratio of the increment of a : the increment 
of m , that is, as the fluxion of a : the fluxion of jjt , by 
Art. 7. 
44. Let y be any number, and * its logarithm ; then if 
x increafe uniformly, or if x be conftant, y w ill increafe 
in geometric progreflion, therefore, by the laft article, y 
y . yx 
varies asj 5 , .-.4 is conflant; hence, — is conllant 5 put, 
y y 
therefore, —=M, and we have i—M x - ; that is, the 
jv y 
fluxion of any logarithm is equal to a confiant quantity 
multiplied into the fluxion of the number divided by 
the number. The quantity M is called the modulus of the 
fyflem, and may be atfumed of any value. 
lfM=i, the logarithms are called hyperbolic, becaufe 
the fame logarithms may be deduced from the hyperbola, 
y 
as will appear hereafter. In tills cafe x—~' 
Prop. XVII. — To find the fluent of a fiuxion, which is the 
fuxion op'any quantity (yJ divided by that, quantity (yJ, or in a 
given ratio to it. 
45. Put the hyperbolic logarithm of y \ then by, 
j * i ^ 
xs — sax' —‘-h 5 -- a2x ‘— 2 —, Sec. — — x 
2 nc r 
s —1 
— 1 x-\-s. - a 2 x"'\ : — 2 .r—, See. the fluent 
2 
of each of which terms is found by the rule in Art. 37. 
hence the fluent required is — x 
71 r r 
S -1 
s. -. a 2 x K + r — 1 
2 
sax mj r’ 
—;—:-;---7--—, See. Now let 
m-\-s ///-fs—1 
us Co. (ider when the fluent of the given fiuxion can be 
exprelfed in finite terms. 
iff. It r, and confequently s, be a whole pofitive num¬ 
ber, the feries ariling from the expanfion of x — a |' will 
terminate, and the fluent can always be found if m be a 
pofitive whole number, or a pofitive or negative fraction. 
Art. 44. -—x, and the fluent of- is x. And as y, al- 
y y 
though here a Ample quantity, mav reprefent any com¬ 
pound quantity whatever, and j its fluxion, we have the 
following 
Rule :—When any fluxional expreflion appears to be 
the fluxion of a quantity divided by the quantity itfelf, 
its fluent will be the hyperbolic logarithm of that 
quantity. 
X 
Ex. 1. The fluent of —j-— is the h. 1 . (hyperbolic lo¬ 
garithm) of x±a. 
2 XX __— 
Ex. 2. The fluent of —7 is the h. \.a 2 -\-x 2 - 
a 2 -\-x 2 
71 X’' -^ X 
Ex. 3. The fluent of--- is the h. 1 . a -px . 
Thefe fluents are obvious, the given fluxion being ma- 
nifeftly the fluxion of the quantity divided by the quan¬ 
tity, for the numerator is the fluxion of the denominator. 
Ex, 
r y 
