FLUXIONS.
40 ;]
b x
Second Form. The fluent of -
vV ±« 2
rithm of X -f- •/x-^a\
Third Form. The fluent of •
bi
is b X hyperbolic loga-
is b x hyperbolic loga-
s/ 2 a x x 1
rithm of a + x -f- f<> a x a 2 .
2 a b x
Fourth Form. The fluent of-■—■— is b X hyperbolic lo-
a — f a? ± x !
x s^a- ^ a-
garithm of •
a 4* V*" a- dt a ;
5. Certain fluents may be found by means of circular arcs of acircle to the radius 1.
Let AD, fig. 15, be a circular arc, put AT, its tangent = t, DBthe sine — y, AB = x, C’l = .r, the radius CD = a, and the arcAD — z\ also let lire tangent D s represent is, D n parallel toAB, s 7i to DB uill express .v andy. Then from the similar tri-
a 3 i ay
angles TAG, DBC, and D n s, we obtain x =
“ 5 + f J f u i
-, and thus four forms of fluxions,V 2 a a — a- VV — a 2
which diflVr only in the signs from those found by logarithms,}uay be found by means of circular arcs, whose lengths are givenhi a table, to a circle,,the radius of which is 1, thus,
bat t
First Form. The fluent of- — b X arc to tang. —.
a 2 i l a
by y
Second Form. The fluent of-= } X arc to sine—.
Third Form. The fluent of ■
S' a*—ifbx
b x arc to vers.
■/ I'm —x 1 0
bus r s
Fourth Form. The fluent of-= b x arc to see. —.
V s l —
T ° the above we add the following, which are easily investigated;the hyperbolic logarithm is to be understood, anti the radius of• the circle is 1.
-ORMS.
Fluxions.
II.
III.
IV.
Fluents.
ii— 1 .
«±c"
i — logarithm of a± x nn
— l
a±x*
An— i
a — x n
An — 1
« + x*
1 xn
— logarithm of-
no. a±^x n
1 fa -{- f x n
■ logarithm of ■
nf a
fa — fx n
1 a — x”
- X arc to cosine --
nfa « + *"
Forms.
V.
VI.
VII.
Fluxions.
Xr)_ 1
l x
f
An— 1
fa— x«
xf2ax—x?
Fluents.
— logarithm of f x n -{- f± a + x»n
• \ X 1 "
arc to sine v —
■ circ. seg. to rad. a and vers, x
(40.) When the given fluxion is so complex that its fluent can-not be found by any of these means, it is then to he determinedby the method of infinite series; thus, expand the radical or frac-tion in the given fluxion by division, extraction of roots, thebinomial theorem, &c. and multiply the series by the fluxionalletter, if il had been left out in the operation, and such variablesimple factor as the given fluxional expression may contain ; thenfind the fluent of each term, by the foregoing rules for the fluentsrequired.
ax
Thus, if it were proposed to find the fluent of -, it be-
a — x
comes necessary to throw the fluxion into an infinite series, by
_ x.v :;-.v
dividing ax by a — x: thus, ax ■—a — x = .i -j- - -(~
a
-j-, Ac. Now the fluent of each term of this series,
i- i 3
may be found by the foregoing rules to be x -|-(-
2a
3a 3
4a 3
-, & c. Again, to approximate the fluent of -
u 2 —xT x x‘x
we first find the value of
r 2 ]!
expressed in a series to be
1 3a
-X .r 2 -\ -
2uc 8 <fi
4 ac 3
8a 3 c
Xi'-j-
oa
16c 7
I
1
I6«c 5
I6« 3 c 3
I6a s c
X x® -}-, &c. which value
being multiplied by x n x, and the fluent taken by the rules above
laid down, wc get
+
3a
8c s
«-f t x cjn+5
a .«+3
«+3
+
5 a
4ac 3 8a 3 c 16 c 7 lCac 5 !6aV i6a 5 c
n+7
+, &c.
(4l.) The fluent found from a given fluxion by the foregoingrules will frequently require a correction, for though a fluent canhave but one fluxion, yet a fluxion may have an infinite number
of