48S 
Ex. 4. The fluent of 
V x 2 - 
FLUXIONS 
is the h. 1. of 
x 4- E x 2 ±:« 2 . For, put x 2 ±:a 2 ~v 2 , then xx—vv, 
x-\-v 
x : v :: v : x, -and x-J-t/ : v :: x-\-v : x ; lienee, 
x-~v v v 
X 
- ■ : therefore the fluent of L lil Z, or of — - 
V x 2 ±a 2 V x ~-\-a 2 ’ 
is the h. 1. x-f-y = h. 1. x-j-V'x 2 ±cz 2 
A 
Ex. 5. The fluent of 
4/x 2 dz 2ax 
is the h. 1. 
xdza-\-f x 2 ± zax. For, put fx 2 ±2ax—y, then x 2 ± 
2axa 2 — y 2 a 2 , and x ± a^Xy 2 -\-a 2 ; hence, x = 
yy xv 
■ - T . confequently y—- = - 7 - whofe 
l/jV -t-« 2 y x 2 ± 2ax V y 2 -\- a 2 ' 
fluent, by the lad example, is h. 1. y-ffy 2 -\-a 2 = h. 1. 
x±a-\-\/ x 2 ztz 2ax 
Ex. 6. The fluent of —■ - is the h. 1 . —For, 
a 2 —x- a —x 
, whofe fluent is the h. 1. a-^x — 
z —x 2 a—x a—x 
{l-~ J~iV 
h.l.a —x=h.l.-, as (hewn in Wood’s Algebra, 
a—x 0 
Art. 388. In like manner, the fluent of 
2 ax 
is h. 1. 
x—a 
Ex. 7. The fluent of — 2a ' x ■ - istheh. 1. ~ ° lit— _f. 
l/« 2 +.v 2 
4/2t~-j- 2 -f-22 
For, put \/ a*-fx*— :v, then a*-]-x*=y', therefore xx—yy f 
2 ox 2ay . 2 ax 2 av 
d 77 = that 1S ’ -=- = whofe 
y x xfa>+x* y 2 —a 2 
fluent, by the laftexample, ish. 1 1, 1 
y + a 
In like manner, the fluent of ——-- is h. 1 . °~ x/a ‘ * 
xf a 2 —x a 
a + 4/ a- —x 2 
becomes > whofe fluent is (by Example 4) 
4 /b 2 J ry 2 
■ h. \.y-'rVb 2 +y 2 h. 1. 1+ 4 /^ 2 + _i_— _ h _ p 
46. Let AD be a circular arc whofe center is C, AT 
its tangent, DB its fine ; draw ins parallel to BD meeting 
the tangent Di in s, and D n parallel to B m. Put CD—a, 
AB—v, BD—4!, AD=z, AT=(, CT—j; then by 
Art. 23. D.= z, Dn—x, ns—j. Now the triangles CBD, 
snE), are fimilar, for they are right angled at B and w, and 
the angle sD/2=CDB, becaufe aDC is the complement of 
ax -7— 
each. Hence, y : a :: x: &=—: butyrzi/CD 2 —CB 2 — 
_ y 
( .. ■ 7 T - - . ax 
4/ a 2 — a —.v 2 =2 4/ 2 ax —x 2 ; .•. z=— - Alfo, 
4/ 2 ax — x 2 
■4 /a\—j 2 ( BC ) : a : :y : 
°y 
Again, by fun. 
4/ a 2 —y 2 
triangles CAT, CBD, s (CT) : a (CA) :: a (CD) : CB; 
Hence, the fluxion of the arc AD, or z, is expreffed un¬ 
der four different forms in terms of the right fine, verfed 
fine, tangent, and fecant ; confequcntiy the fluent of each 
of thefe fluxions will be expreffed by z. Hence, 
1.-Fluent 0^ — - 7 L 
is a cir. arc whofe rad. is a and 
4/ a 2 —_y 2 
fine y. 2. Fluent of 
4/ 2 ax —x 2 
; is a cir. arc whofe rad. 
is a and verfed fine x. 3. Fluent of 
a 2 t 
a 2 -ft 2 
is 
cir. 
Ex. 8. The fluent of is —h. 1 
4/ b* J \-x 2 x 
For, put - =y, then x~ 2 x= —y ■ hence, the fluxion 
1-ff l-\-b2X 2 
X ' 
Tliefe are the moll ufeful forms of fluxions whofe 
fluents may be found by a table of hyberbolic logarithms • 
which table may be fnpplied, by multiplying the loga¬ 
rithm found from the common tables by 2,30258509, 
which will give the correfponding hyperbolic logarithm. 
£ _ 
Ex. The fluent of ——is the h. 1 . of i-J-x; if x 
1 ~\~x 
—1, the fluent is the h. 1. of 2—0,693147 ; if x—4, the 
fluent is the h. 1. of 5=1,6094379. 
To find FLUENTS by CIRCULAR ARCS. 
Prop. XVIII .—The length of a circular arc for every de¬ 
gree, minute , and fecond, to radius = 1, being given, to fad 
from, thence certain fluents. 
whofe rad. is a and tangent t. 4. Fluent of — ■ ■ 
sfs 2 — a 2 
is a cir. arc whofe rad. is a and fecant s. 
Now by a table exhibiting the length of circular arcs 
for all degrees, &c. of the quadrant to radius unity, if 
thefe arcs be multiplied by a we fhall liave their lengths 
to the radius a. Hence, for example, what is the fluent of 
ay 
— ■ . when y is the fine of 30"? The length of an 
4/ a 2 — y 2 
arc of 30° to radius 1, is 0,5235987 : hence, the length of 
the arc to radius a, is .a x 0,5235987, the fluent required. 
Thus, the fluents of all fluxions under any of thefe 
forms may be found. 
47. A fluent can have but one fluxion, but a fluxion 
may have an infinite number of fluents; thus, the fluent 
of x is x, or x±a, whatever be the value of the conftant 
part«. By Prop. 4. in taking the fluxion of a binomial, 
the conftant part goes out, and therefore when the fluent 
is taken back again, that conftant part does not appear. 
Now to determine, in any particular cafe, what this con¬ 
ftant part is to be, or whether any Inch quantity is to be 
-annexed, confider whether the fluent firft taken becomes 
equal to nothing, or of a known value, at the time it 
ought; if it do, it requires no conftant quantity to be 
added ; if it do not, fuch a quantity muft be annexed to 
it, as will make it bei-Oine equal to nothing, or to its pro¬ 
per value. This iscalled the correElion of a fluent. 
48. Although the fluxion of a quantity be relative , 
that is, if x denote the fluxion of x, then will wx“— i x be 
the fluxion of x", where x'may be aflumed of any magni- 
1 tude, 
