48(5 
FLUXIONS. 
Ex. 3. What 5 s the cube root of a —z, or the value of 
a —z) in a feries ? Firft, a — z^—a 3 X 1 — - j and com- 
TIi - z 1 1 a 
paring 1 — - with 1 —a ) , we have - —x, n— - hence, a 3 
til a _ 3 _ 
1 h— 1 2 z 2_ . 
1 1 z i 4 —x 
X I- =«3 X I-- 
a 1 3 a 3 2 
&c. 
3<2T 9123 8 xa 3 
Ex. 4. What is the value of 
in an infinite 
feries ? Firft, 
V az—z 2 
1 
V az—Z 2 A 1 
a 2 z 2 X 
_i. i ( ri 
j_fl 2 ; and comparing 1—J 2 with 1— x] we have 
a al 
z 1 , 1 ' 7 ]—b 1 
-=*> n— -; hence, -p-pX 1-=-TT X 
2 a 2 z 2 a a?z' 2 
— 1 2 , • 
1 -.- . 
2 a 
-1 z^ 
-I 77 — 1 —4—2 z° 
2 « 2 
3 * 
• 7 ? +> 
1 x ■ a a. ■ z 
a 2 z 2 2a 2 8 a 2 16a 2 
+> &C. 
Ex. 5. To rcfolve 
<2 2 + 2aX + X 2 
I _ 
into an infinite feries. 
This quantity is p— a-\-x) ; which compared 
with a- |-a|", gives k =2 — 2; hence, a-|-x]— 2 = a 2 — 
- 2—1 _ —2—1 —2—2 _ 
2 a — 3 x —2.--. & 4 * 2 —2.-.-. a 5 ^ 3 —. 
1 ix 1.x 2 4X 3 . c 
& c- — TS “ 75 + -7T+> &c ” 
a 2 a 
Ex. 6. What is the value of 
zaz-\-z 2 
in an infinite 
feries. This quanfity is equal to 
=-X 
2 az 
zaz x1H- 
2 a 
with 1-1-2.1' > 
2 JZ 
jft— : and by comparing 1+ — 
zu * ' 2x2 
we have x =2— « —— 1 • hence, x i-l— = 
za zaz zal 
1 Z -1-1 z 2 
-X'-l,-!•-.-2 
zaz za 2 4a 2 
-, See. 
zaz 4a 2 
-, Sec. In like manner we mud proceed in the 
8a 3 
expanfion and divifion of all binomial quantities. 
35. The value of 14-x)" has been affumed —\-\-ax-\- 
lx 2 fcx 3 -\-, See. and applied in all cafes, whether n be a 
whole number or a fraction; if « be a whole number, it 
is manifeft from the obfervation in Art. 33, that this 
muft be the form of the feries; but if n be a fraction, 
it is not fo obvious that we may affume the fame feries; 
the legality of the aflumption however in that cafe may 
7* 
be thus fhewn. Let n— any fraction r and s being 
whole numbers. Now the value of I"+w| is exprefted 
by i-\-ax-\-bx 2 -\-cx :l -\-. See. but j+xK is the s th power of 
1 +*) 7 , therefore fuch a feries muft be affumed for 1 -fx ]7 
that the s th power thereof may give a feries of the form 
tfax-\-bx 2 f-cx 3 -\- } Sec. Now any power of the feries 
i+px+bx 2 -\-rx*-\-, Sec. will give a feries i-f ax-J-x& 2 -}-o;S 
+, &c. therefore we may affume a feries of that form, 
where the powers of x regularly afeend, to reprefent the 
value of i-J-xl'' 
METHOD OF FINDING FLUENTS. 
36. The bufinefs of the direSl method of fluxions is to 
find the fluxion from the fluent ; to find the fluent from 
the fluxion is fometimes called the inverje method of 
fluxions. It is not difficult to put any quantity into 
fluxions, there being direft rules for that purpofe; but 
there are no direct general rules for finding a fluent from, 
a fluxion; and very often it is impoffible to do it, ex¬ 
cept by an approximation by an infinite feries, as the 
fluxion may be fuch as could not arife from putting any 
fluent into fluxions. We cannot therefore lay down rules 
for finding the fluents of any other fi ixions than thofe 
whofe forms ffiew them to have been derived from f'ome 
fluent. 
Prop. XIV. — To find, the fuent of any power of a fmple 
quantity multiplied by the fuxion of that quantity. 
37. The fluxion of x 3 is 3x 2 x, therefore we know that 
the fluent of 3 v 2 x is x 3 , and it is deduced from the fluxion, 
by the converfe of the rule for putting x 3 into fluxions. 
In general, the fluxion of x" is, by Art. 12. nx' — ‘i; 
therefore the fluent of nx”— 1 x muft be x", and this fluent 
is deduced from the fluxion by the following 
Rule:—A dd unity to the index, divide by the index 
fo increafed, and alfo by the fluxion of the root. 
Ex. 1. The fluent of qx 6 x is x 7 . 
x i o 
Ex. 2. The fluent of x 9 x- is -—• 
10 
Ex. 3. The fluent of $x 3 x is -— 
4 
Ex. 4. The fluent of -xTx is -X-X*^=—^ 
9 8 9 2 4 
Ex. 5. The fluent of or 6.v 9 .vis —-=-L. 
x 9 —8 4x 8 
Ex. 6. The fluent of or 3 y —fj is-X 3 j 4 =— yi 
y-f 2 ' 3 z J ’ 
38. If n— o, or the index of x be —1, the fluxion is 
x 
~; but this fluxion cannot be generated by x°, becaufe 
(by the principles of Algebra) x°zzi, a conftant quantity ; 
X 
hence, the fluent ofk cannot be found by this rule. 
x 
Prop. XV. —To find the fuent of a binomial quantity (one 
part of which is corf ant and the other part variable J raifed to 
to a power, where the term without the vinculum is the fuxion 
of the variable tern under the vinculum, or in a given ratio 
to it. 
39. The fluxion of a'-\-x'l" is (by Cor. Art. 12.) n X. 
a r f-x\" — *X.rx r — 1 x, which is found by the fame rule as 
tlie fluxion of x". Every complete fluxion, therefore, of 
this kind muft neceHarily have the index of the variable' 
quantity without the vinculum, lefs by unity than the in¬ 
dex under the vinculum. Hence, every quantity fo circum- 
ftanced may have its fluent found by the above rule. 
If r— 1, then r— \—o, and x°=i ; theretore the fluxion 
becomes r.yfa-\-x j” 1 y.x. 
Ex. x. What is the fluent of a+x] 6 Xx ? Here the 
fluxion of the root afx is x ; hen c e the fluent is 
a-\-x\ 7 y.x _ afx) 7 ' 
•j x 1 j 
Ex. 2. What is the fluent of a 2 -\-x*) 7 Xx& ? Here the 
fluxion of the root a 2 -|-x- 2 is zxx ; hence, the fluent is 
a 2 -\-x 2 1| x xx_ a 2 -{-x -12 
I-X2X.V — 3 _5 
Ex. 3. What is the fluent of a 4 — ? ^ ere 
tl'.e 
