Isaac Newton.

# The method of fluxions and infinite series : with its application to the geometry of curve-lines online

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tion is now 4 = j- -|- - , or putting x = i, it is aby -f- axj
/y ~+- yx -f- a 1 , which may be thus refolved :

aby =

xy

\

ab a* 2/1*
/7* .. V 1 ,

4. fc at> ^ /34

-i n *b ab 1 6,3

c__

b

- 4- "*x -f-

ab aa

babl v J

OCC.

^x.

>

b

a I ,

2.U 3 J

/?.6 ^ ifi h^

OCC.
c.

(IX -(

ib * +

" v-J-

i}ab~ f
a I,

OtC.

)

a b a z* J -

b A "^

"*-"*.., , I' +2.

,V, ^ J 6^3

OCC.

b 4- h ; ^ -f- 2 ,

a b a -iri' -L-

i ""

OCC.

rt

Arr

2 Difpofing

and INFINITE SERIES. 287

Difpoiing the Terms as you fee is done here, make a 1 - the firft
Term of aby, then ~ will be the firft Term of j, and thence -|x
will be the firft Term of y. So that a x will be the firft Term of

* b

axv, and ax will be the firfl Term of by. Thefe two to-
/ * i i /

gether, or -,x ax = - x, with a contrary Sign, mud be put
down for the fecond Term of aby. Therefore the fecond Term of
y will be '-~-x, and the like Term of y will be A*. Then the

' t>- -' ^b-

fecond Term of ..\y will be '-'-~ a A*, and the fecond Term of

b"

ab

ly will be -^~ x *, and the firft Term of xy will be yA*.
Thefe three together make - ~ 2 * _ ~ A*, which with a contrary
Sign muft be made the third Term of aby. Therefore the third
Term of y will be ~-^-'A* and the third Term of y will be

r ZaL ' *

t , / * _^_ 7

- A' 3 . And fo on. Here in' a particular cafe if we make.

b =. a, we mall have the fimple Series y =. x * -+- ^ 7 , &c.

Or if we would have a defcending Series for the Root y of this
Equation, we may proceed as follows :

xy~\ =<!* a ~f- b x a**- 1 -+- zu\ +- zab -+- i x a*x-*, &c.
[,y\ - - _f_ a*bx~ l a-i-l) y.a ! -bx~-, &c,

~' -~ t , &c.

'-, &c.
3 6cc.

JJ/=: rt a A~ ; - ^ -4-/>X 2rt*A-~ 3 , &C.

Difpofe the Terms as you fee, and make a* the firft Term of the
Series xy, then will - be the firft Term of y, and a*x~"- will

be the firft Term of y. Then will -f- a"-bx~ l be the firft Term of
by, and a"'X~ I will be the firft Term of axy y which together
make a-\- b x a l x~ t ; this therefore with a contrary Sign muft be
the fecond Term of xy. Then the fecond Term of y will
be a-\-by.a 3i x~~ ) and the fecond Term ofj/ will be a-\- l>-x.2a 1 x~ 3 ,

Therefore the fecond Term of by will be a -f- b x tf*Av~*,

and

2 88 TZe Method of FLUXIONS,

and the fecond Term of axy will be a-\-b* 2a*x~*, and tlie
firft Term of aby will be a^bx~- ; which three together make
^za 1 -I- zab -+- b* xa 1 *.*. ThiswithacontrarvSisnmuftbethethird

Term of xy, which will give 2a* -+- zab -+- b % x a 2 x~ 3 for the
third Term of y ; and fo on. Here if we make b=.a, thenj=

a 1 za* ca4 .

- +- r ^T 3 &C.

x *' x* '

And thefe are all the Series, by which the value of y can be ex-
hibited in this Equation, as may be proved by the Parallelogram.
For that Method may be extended to thefe Fluxional Equations, as
well as to Algebraical or Fluential Equations. To reduce thefe
Equations within the Limits of that Rule, we are to confider, that
as Ax m may reprefent the initial Term of the Root jr, in both thefe
kinds of Equations, or becaufe it may be y = Ax m , &c. fo in
Fluxional Equations (making #=1, we mall have a\foy=mAx m ~ I )
6cc. or writing y for Ax m , 6cc. 'tis y = myx~* t , &c. So that in
every Term of the given Equation, in which y occurs, or the Fluxion
of the Relate Quantity, we may conceive it to take away one Di-
menfion from the Correlate Quantity, fuppofe x, and to add it to
the Relate Quantity, fuppofe y ; according to which Reduction we
may inlert the Terms in the Parallelogram. And we are to make
a like Reduction for all the Powers of the Fluxion of the Relate
Quantity. This will bring all Fluxional Equations to the Cafe of
Algebraic Equations, the Refolution of which has been fo amply
treated of before.

Thus in the prefent Equation aby -+- axy = by -f- yx +- aa, the
Terms mufl be inferted in the Parallelogram, as if yx~ ' were fub-
ftituted inftead of y ; fo that the Indices will ftand as in the Margin,
and the Ruler will give only two Cafes of exter-
nal Terms. Or rather, if we would reduce this
Equation to the form of a double Arithmetical
Scale, as explain'ci before, we mould have it in this
form. Here in the firft Column are contain'd thofe
Terms which have y of one Dimenfion, or what _ y i_
is equivalent to it. In the fecond Column is a 1 , + ax i ; J 2 C~
or y of no Dimenfions. Alfo in the firft Line is
. xy, or fuch Terms in which x is of one Dimenfion. In the,

fecond Line are the Terms by~l , . ,

<f a 1 , which have no Dimen-

{iqiis of .v, becaufe -j- axy is regarded as if it were ay. Laftly,
in the third line is aby t or the Term in which x is of one negative

Dimenlion

p-

2.V + Xj~

and INFINITE SERIES. 289

Dimension, becaufe -\-aly is confider'd as if it were -f- abx~~ J )\ And
thefe Terms being thus dilpos'd, it is plain there can be but two Cafes
of external Terms, which we have already difcufs'd.

?2. If the oropofed Equation be = -TV 2 x -4- or

O jy y xx >

making x= i, 'tis y -f- 3_v 2.v -f- xy~ l 2yx~ 1 = o ; the
Solution of which we mall attempt without any preparation, or
without any new interpretation of the Quantities. Firft, the Terms
are to be difpos'd according to a double Arithmetical Scale, the Roots
of which are y and .Y, and then they will Itand as in the Margin. The
Method of doing this with certainty
in all cafes is as follows. I obferve in
the Equation there are three powers of 1
y, which are y 1 , y, and 7-' ; there- *
fore I place thefe in order at the top
of the Table. I obferve likewife that there are four Powers of x,
which are .v 1 , x, A I , and x~ l , which I place in order in a Column
at the right hand ; or it will be enough to conceive this to be done.
Then I infert every Term of the Equation in its proper place, ac-
cording to its Dimenfions of y and x in that Term ; filling up the
vacancies with Aflerifms, to denote the abfence of the Terms be-
longing to them. The Term y I infert as if it were _}'*""',
as is explain'd before. Then we may perceive, that if we apply the
Rukr to the exterior Terms, we mail have three cafes that may pro-
duce Series ; for the fourth cafe, which is that of direft afcent or
defcent, is always to be omitted, as never affording any Series. To
begin with the defcending Series, which will arife from the two
external Terms 2x and -f- xy~ s . The Terms are to bsdifpos'd,
and the Analyfis to be performed, as here follows :

- 2JX-

-J44*-*, &c.

Make xy~ l = 2X, 6cc. then y- 1 = 2, &c. and by Divifion
T=4, &c. Therefore 3>'=T, &c. and confequently A-> '=
" 4, &c. or y- 1 = # -I*" 1 , &c. and by Divifion y = * -f-
-f.*- 1 , &c. Therefore 2)' = *^ x ~ 1 y & c - an d confequently x\~ l
r^ % * T*" 1 ) &c. So that y~ l = * * T*"~% ^ c ' anc ^ ^7 ^' v ^"
fion y = * * -f- - r v f x~'^c . Then 3v = * # -j- 4r A ~% ^ c - anc ^

P p y

Method of FLUXIONS,

y == * -f- i*-"-, &c. and zyx~* = A % 6cc. Thefe three-
together make 4- r^x- i , and therefore xy~ l = * * * 44*"%
&c. fo that y * * * -f- V|T*~~ J & c - A "d fo on.

Another defcending Series will arife from the two external Terms
-4- -y and 2X, which may be thus extracted :

zx f -f- 41*-' i|*- 3 , &c.
' _ ^4x-*, &c.

+^ X -, &C.

i-x-% &c.

Make 3/ = 2X, &c. then y = ^x, &c. and (by Divifion) y *
= x~*, &c. and x>'~ 1 =|,&c. and y=- T> & c - There-
fore 3_y = * , &c. and _y = * T S T) &c. and (by Divifion)
xy~* = * -g-*" 1 , &c. and _y= * o, &c. and zyx~* =
~x~\ &c. Therefore 3_y=* * 4- i-j..*- 1 , &c. and jy = * * 4-
^.i^ J , 6cc. &c.

The afcending Series in this Equation will arife from the two ex-
ternal Terms 2yx~* and xy~ l ; or multiplying the whole Equa-
tion by y, (that one of the external Terms may be clear'd from
y,) we mall have yy 3^* 4- zxy x 4- 2y i x~ 1 = o, of which
the Refolution is thus :

- v \ S v* -1 vl- _I_ 9 - v3 &r

M^~ t^V # ^S~" ^^* ^ . - "T"*\ LA/V *

r / ^ *r '

Jl

2 4

6cc.

; :*

v 2

&c.

^a J_ ' 35 A

&c.

y

y __ 3 ^.I # 1^.^ 3^^,

Make aj 1 ^- 2 = ^, &c. then y* = 4-AT 3 , &c. and y = x
Here becaufe of the fractional Indices, and that the firft Term of
4- kxy, or 4 x% } may be afterwards admitted, we mufl take o

for the fecond Term of 2)-A % and therefore for the fecond Term

i of

and INFINITE SERIES. 291

of y. Then y'y = **> &c. and confequently 2v*x~ a = * * A* 1 ,
8cc. and y 1 = * * 4-v*, &c. and by extracting the fquare-roor,

Then yy = -f- o, &c. and 2.vy = -4 - V-'

&c. and therefore 2)' l x~ t = * * -^^S &c. and _>' = * * *

|.v 5 , &c. &c.

33, 34. The Author's Procefs of Refolution, in this and the fol-
lowing Examples, is very natural, fimple, and intelligible; it pro-
ceeds Jeriatim & terminatim, by p'afling from Series to Series, and
by gathering Term after Term, in a kind of circulating manner, of
which Method we have had frequent inftances before. By this
means he collects into a Series what he calls the Sum, which Sum

is the value of - or of the Ratio of the Fluxions of the Relate

,v

and Correlate in the given Equation ; and then by the former Pro-
blem he obtains the value of y. When I firft obferved this Method
of Solution, in this Treadle of our Author's, I confefs I was not u
little pleafed ; it being nearly the fame, and differing only in a few
circumftances that are not material, from the Method I had hap-
pen'd to fall into feveral years before, for the Solution of Algebraical
and Fluxional Equations. This Method I have generally purfued in
the courfe of this work, and fliall continue to explain it farther by
the following Examples.

The Equation of this Example i 3^ -f- y + x l -+- xj y
= o being reduced to the form of a double Arithmetical Scale,
will (land as here in the Margin ; and the v , v<)

Ruler will difcover two cafes to be try'd, of ~
which one may give us an afcending, and the xI
other a defcending Series for the Root y. And
firft for the afcending Series.

The Terms being difpofed as you fee, makej/=i, &c. then
y=x, &c. Therefore y = x, &c. the Sign of which Term
being changed, it will bej/= * -{- x 3 AT, &c. = * 2.v, &c.

P p 2 and

292 77->e Method of FLUXIONS,

and therefore y = * x x , &c. Then y = * -+- #% &c. and
.vy = *% &c. thefe deftroying each other, 'tis y = * * -+**,
&c. and therefore _y = * *-t-7.* 3 , &c. Then _y=** ^x*,
&c. and xy = * -f- x', &c. it will be j- = * * * .I* 5 , &c.
und therefore y = * * * ^x* y &c. &c.
The Analyfis in the fecond cafe will be thus :

h x 4 ~t-

V =

AT

4

* -+ I2X~ 3 , &C.

6*- 1 * , & c .

f~ 2 1 2X~1, &C.

6*-" 1 +- 6.V~ Z * I2AT~*, 6CC.

Make xy = x l , &c. then ;' = x, &c. Therefore _y
= x, &c. and changing the Sign, 'tis xy-=. % x 3*, &c.
= * 4*, &c. and therefore y = * -h 4, &c. Then jy= *
4, &c. andj = i, &c. and changing the Signs, 'tis xy
= * * H- 5 -f- i, 6cc. = * # -l- 6, &c. and y = * * 6x~*,
&cc. &c.

35, 36. If the given Equation were ^ == :i-f-^-f.^._f_^- y
H ^ , &c. its Refolution may be thus perform'd :

zx'i

XV

a*

a
X

A*

A 1 *

&c -

4 > & c -

, &c.
1, &c.
- , &c.

A*

A4

f *" * 2^2 *^ ^^4 "T~ i/)3 "I" o>,4

Make y
&c. and y

i, &.c. then y= x, &c. Therefore 2 f

o a

* + ;, &c. and therefore _y =*+-, &c. Then

therefore

y =

* *

~i = J7 j
, &c. and j = * * + , &c. And fo on.

Now

and INFINITE SERIES. 293

Now in this Example, becaufe the Series | -+- ^ 4- ^ -+.
. &c. is equal to =? it will be y= h I, or ay . xy

4 ' a x ' / A; ' J

=jy + tf *", that is, jx -f- rfx xx ay -+- A j = o ; which
Equation, by the particular Solution before deliver'd, will give the
relation of the Fluents yx ay -{-ax I** = o. Hence y =
a * -_xx an( j , Divifion y= x -f- * h -, -f- r , &c. as found

a x J za za~ 2a* '

above.

37. The Equation
of this Example being
tabulated, or reduced
to a double Arithmeti-
cal Scale, will ftand as
here in the Margin.
Where it may be ob-
ferved, that becaufe of
the Series proceeding both ways ad injinitum, there can be but one
cafe of exterior Terms, of which the Solution here follows:

x-'

>

1

x

(

*

~H j 1 *4" j* *4* ^* > ^ f -

X*

3*; 4-

3 xv

X)! 1 jyJ Xj4,&c.

X 1

6# 1 j'

* * *

xt

8*3 +

8*5,

* * *

'= O.

Jt4

1 ox 4 -f-

0*4y

* * *

X*

I 2X* -j-

z.'. '_y

& f .

1 4 * <

5^f.

= , A; 6x l

iox+ i2x s i4.x s , &c.

-f- |

7 3

-

-f-

c s , &c.

Y 6 &C

\ j CW .

X4 6x-' -4- 1 * 6 , &c.

s , &c.

X 6 , &C.
X 6 , &C.

- 4- * 6 > &c.

v . i 2%l , X 4 ^'Y' Y 15 ! . 3&7 ,V 7 &C.

y*

&c.

Make y = 3^-, &c. then y = 4x a , &c. Then y = *
6x*, &c. and_y == # 2 A' 3 , &c. Then 3^ = -f- |x3, &c. and
therefore j= * * x 3 8^ 3 , &c. = * * "V x 3 , 6cc. and _y
= * * VAT*, &c. And fo of the reft.

The Author here takes notice, that as the value of y is negative,
and therefore contrary to that of x, it fhews that as x increaies, /
muft decreafe, and on the contrary. For a negative Velocity is a
Velocity backwarks, or whole direction is contrary to that which

was

294 Th* Method of FLUXIONS,

was fuppos'J to be an affirmative Velocity. This Remark mull take
place hereafter, as often as there is occafion for it.

38. In this Example the Author puts x to reprefent the Relate
Quantity, or the Root to be extracted, and y to reprefent the Cor-
relate. Bat to prevent the confufion of Ideas, we mall here change
.v into y, and / into A", fo that y (hall denote the Relate, and x the
Correlate Quantity, as ufual. Let the given Equation there-fare be

- = ~x 4** -\- 2xy' i -f^ 1 -h 7#* -+- zx' } whofe Root y is to

be extracted. Thefe Terms being difpoled in a Table, will ftand
thus: And the Refolution will be as follows, taking y and -t- x
for the two external Terms.

X 1
X 1

a
* x

X 1

At

. 1

* * * + ** 5 J I _j_ ^i 2J34-4* 1

* * '* _|- 7A; i: If -t Z-

J = I*' 1 * - A'H-ZX + I*

** -

* * *

4..1

x/ # * *

* * * *

* *

y

j/=|A;, &c. then^ = -l-x a , &c. Now becaufe it is jx=s
* o, &c. it will be alfo y= * o, &c. And whereas it is^ = |-x,
&c. it will be zxy^ = x*, &c. and therefore y = * # -f- x*

4**, &c. = * 3**, &c. then _y= * * >r 3 , &c. Now be-
caufe it is y = * -f- o, &c. it will be alfo y^ == * -\- o, &c. and

2Ay 5 = * -f- o, &c. and confequently y E= ***-{- 7^^, &c. and
therefore y = ***-{_ 2**, &c. And fo on.

There are two other cafes of external Terms, which will fupply
us with two other Series for the Root y, but they will run too much
into Surds. This may be fufficient to (hew the univerfality of the
Method, and how we are to proceed in like cafes.

39. The Author mews here, that the fame Fluxional Equation
may often afford a great variety of Series for the Root, according as
we fhall introduce any conftant quantity at pleafure. Thus the
Equation of Art. .34. or j/=i 3* -\-y -j- # -f- xy, may be re-
folved after the following general manner:

and INFINITE SERIES.

295

r^3*+ ** y = + * * l

a 4. x -fza* 1 + i<?*', <:. + ax+ax 1

Ji ax ax 1 ^
ax x* ,

axt, isV.

Here inftead of making ji/ = i, 6cc. we may make _y=o, &c.
and therefore y = a, &c. becaufe then y = o, &c. then y
a, 6cc. and confequently y = * -f- <z -+- i, &c. and therefore^ 1
=; * -+- ax - x, 6cc. Then y = * ax x, &c. and xy
== _- ##, &c. and therefore y = * * -f- zax -f- x 3*, &c. =
* * -f- 2ax zx, &c. and then y = * * -f- ax* x 1 , &c. There-
fore y = * * ax 1 -f- x 1 , 6cc. and xy= * ax* AT*,
&C. and confequently y = * * * -f. tax* -f- x*, 6cc. and y =
a, * * -f- .iflx 5 -f- -i-* 3 , &c. &c. Here if we make a = o, we fhall
have the fame value of y as was extracted before. And by what-
ever Number a is interpreted, fo many different Series we fhall
obtain for y.

40. The Author here enumerates three cafes, when an arbitrary
Number mould be affumed, if it can be done, for the firft Term of
the Root. Firft, when in the given Equation the Root is affected
with a Fractional Dimenfion, or when fome Root of it is to be ex-
tracted ; for then it is convenient to have Unity for the firft Term,
or fome other Number whofe Root may be extracted without aSurd,
if fuch Number does not offer itfelf of its own accord. As in the
fourth Example > tisA' = i}' 1 , &c. and therefore we may eafily have

x^ = -i->'> & c> Secondly, it muft be done, when by reafon of the
fquare-root of a negative Quantity, we fhould otherwife fall upon
impoflible Numbers. Laftly, we muft aflame fuch a Number, when
otherwife there would be no initial Quantity, from whence to begin
the computation of the Root ; that is, when the Relate Quantity,
or its Fluxion, affects all the Terms of the Equation.

41,42,43. The Author's Compendiums of Extraction- are very
curious, and fhevv the univerfality of his Method. As his feveral
ProcciTes want no explanation, I lhall proceed to refolve his Exam-
ples by the. foregoing general Method. As if the given Equation

werej=:- x 1 , or y /-' = x 4 , the Refolu-tion might
be thus :

y

296 The Method of FLUXIONS,

'y T = O * * < . X* l<3-7x 3 , &c.

f 1 f ' a.~ I -f- a~*x \a~\$x*- -\- a~ 7 x*, &c.

J 4-^~*A" ;

/7 _1_ * ** _I_ 'IL 1^ AT

-t- - j -4- ,77 g a , , see.

Make _y = o, &c. then afluming any conftant quantity a, it may
be y= a, &c. 'Then by Divifion y~ l = > a~ l , &c. and
therefore _y = * -f- a* 1 , &c. and confequently _y = * -4- a~ l x, 6cc.
Then by Divifion y~ l = * -{- -3x, &c. and therefore y =

* * a~ix, &c. and confequently _y = * * 4~ 3 ^S & c - Then
again by Divifion y-' 1 = 4^ s'x 1 , &c. and therefore y =

* * *H-|d~5 A ; 1 .vv&c.and confequently/ = * * * a-*x* ^x',
&c. And fo of the reft. Here if we make a = i, we ihall
have y = i -f- x IK* -f- .x* |-|-Ar 4 , &c.

Or the fame Equation may be thus refolved :

- y~ ' J == - A" 1 -f- 2 AT" 3 -f- I4AT- 8 -+- 2 l6x- J 3, &C.

8 ' 2i6x'~~ I 3, &c.

= AT" 2 -f- 2AT~ 7 + l8x- JZ + 28ox- I 7 ) &C,

Make y~'= A-S&c. or_y=^~ z , &c. Thenj/= 2Ar~ 3 ,
2fc.and therefore y l =z* -\-2x~ 3 , &cc. and confequently by Divifion
r=* -f-2 .v~ 7 , &c. Then j/=* i4x~ 8 ,&c. and therefore y- 1
= * *4-i4.v~ 8 , &c. and by Divifion _)'= * *+i8^^ 12 , &c. Then
y = * * 21 6jf J 3 3 &c. and therefore y~ l = * * * -f- 2 l6x~ l ^ }
&c. and by Divifion y = * * * + 28o^~ 17 , &c. And fo on.

Another afcending Series may be had from this Equation, viz.

y=^/2x \ X' -f- ** -f- ^ , &c. by multipying it by y, and

then making i the firft Term of yj.

44. The Equation y = 3 -+- 2y x~ J y- may be thus refolved :

-4- ojc- 1 , &c.

r

y y

-|- 3-x 11 , &c. ^1
9x a , &c. x~ l
l -{- ,?x 3 , 6cc.

*

+27+3 ? :=:0>

' a JV * J

Make

IN FINITE SERIES. 297

Ma ke y = 3 ,Scc. then y = ^x, 6cc. Therefore zy = 6x, &c.
and x~ I y i = c)x, &c. and confequently_>' = * 3*, 6cc. Therefore
y == * IA*, &c. Then a_>' = * -f- 3* 1 , 6cc. and x~ I y l = ==
* 9**, 6cc. Therefore^ = * * -f- 6# 4 , &c. and / = * * -f- 2AT 3 ,
&c. &c.

Or the Refolution may be perform'd after thefe two following
manners :

zy 1=3 *-' -f- IA *,&c. ;'*-'\=

*

_? v~~ l 1 -* v~~ '4 ' v~"~ 2 &"f*

_j _/ ^ ^^^r.-v 4 "F^ j A'^*

T = 2A - j-i 3 A-~' &C.

Make zy-=. 3, &c. or_y =r ?, 6cc. then j= o, &c. and
x -iy - =-j- %x- 1 , &c. Therefore 2_y = * %x~ l , &c. or /
= * -f- T*"", 1 } &c. and y = * % x ~ z > & c - anc l by fquaring x~'j*
= * I 1 -'''" 2 , &c. and therefore 2_>'=r* * -f- ^A*~ 2 , 6cc. and
y = * * I-*" 2 } &c. And fo on.

Again, divide the whole Equation by y, and make x J y = 2, &c.
thenj' = 2A;, &c. And becaufe j/=2, &c. and j'^ni^ 1 " 1 , &c.
'tis^" 1 = Aj 1 "" 1 , &c. and 3>'~~ 1 = I-* 1 "" 1 } ^ c - therefore yx~*
= * H- T-^" 1 ) ^ C - an( l y = * H- T> & c - Then becaufe j^y" 1 ==
* + o, &c. and y I = * -+- -I-*" 2 , & c - 'tis jx~ l = * * T" v ~ 4 >
&c. and y * * T X ~*> ^ cc> ^ c -

45, 46. If the propofed Equation be_y = y -\- x~' X~-, its
Solution may be thus :

77 = x-*+x~

s

4-jrJ _!_*-

/;

)=A ' A'" 1

y' y

^

(

> -f-A" *

x

y * )

+^:

! - ' X~ z

AT" 1

j -f-A-~'>

*>

A 1

* x ~'j

Makejj/= x~ z , &c. then y =r A ', &c. Confequently }'=*.
* o, &c. and therefore _y= * o, &c. that is, y = x~ I .

Again, make y^ix" 1 , &c. then y =. A-~% 6cc. and confe-
quently _>' = * + o, &c. that is, y = A'.

That this fhould be fo, may appear by the direcl Method. For
ify = x~ l , 'tisj/ = A-A-~ 2 ; a\foyx= .\x~'. Then adding tlieie
two Equations together, 'tis yx -\-y =xx~ ' xx~*, orj = y

x~ l x~*. Thus may we form as many Fluxional Equations

as

298 The Method of FLUXIONS,

as we pleafe, of which the Fluents may be exprefs'd in finite Terms;
dients. Thus if we aflume the Equation y = 2x x* -f- ^ } ,
taking the Fluxions, and putting x= i, we {hall have jr = 2

|x -f- -for 1 , as alfo ~ = i ^x -+- ~x*. Subtract this laft from
the foregoing Equation, and we fhall have j/ = i 2#-f-l* t ,

ZX

the Solution of which here follows.

47. Let the propos'd Equation be y= -f- i , 2#-{- !#*, of
which the Solution may be thus :

\ e fx gx* __ ex' 1 fx

'

= o.

By tabulating the Terms of this Equation, as ufual, it may be
obferved, that one of the external Terms y -+- ^yx~ l is a double
Term, to which the other external Term i belongs in common.

Therefore to feparate thefe, afllime y = zex, &c. then . i-

= e, &c. and confequently y = i -f- e , &c. and therefore y
==x -f- ex, &c. That is, becaule 2ex = x -f- ex, or ze=. i -j-e,
'tis ez= i, or _y= 2x, &c. So if we make _y=* -f- 2/x z , 5cc.

then i- = * y^j &c. therefore y = * ~\~fx 2x, &c. and

v = * -+- l/x 1 A-*, &c. that is, 2/= |/ i, or /= -i. So
that _y == * -f-A'% &c. So if we makejv' = * * +- zgx*, &c. then

s== * * gx 1 , &c. and therefore jj/ = * * -+-gx* -4- ix 1 , <Scc.
and/ = * * -f- |*-> -f- ^s &c. or 2g = -jg+, or ^ -_ __. > fo
that j =# * ^-.v 3 , &c. So if we make / = * *. * 2/^x 4 , &c. then

-=*#*- /w 3 , &c. and therefore _y= * * * -f- /JA: 3 , &c.

and /= * * * -f- ^r^ 4 , &c. Butbecaufe here 2^=^^, this Equa-
tion would be ubfurd except = o. And fo all the fubfequent Terms
will vanifh in infinitum, and this will be the exact value of y. And
the fame may be done from the other cafe of external Terms, as

48. Nothing can be added to illuflrate this Investigation, unlefs
we would demonftrate it fynthctically. Becaufe^ =ex* } as is here

found,

and INFINITE SERIES. 299

found, therefore y = e: v+~', or y = Iff! . Here in (lead of ex' (

fubftitute y, and we fhall have_y = ~ x , as given at firft.

49, 50. The given Equation y =yx~- 4- x~- -f- 3 4- 2.V 4*-'
may be thus reiblved after a general manner.

y n = 2x 4- 3 4A- 1 4- x- 2 .v-* 4- i-*-* , &c.
/ -f- i -f- 4-v ' 4-rf.v~ I rtx~~ 3 4- ax~*

-~ x ~ z )\ - - - * 4 A "~' a *~* + *~ 3 T*~* , &c.
J -\-ax~* ax~*

y= x l -f- 4.v + a #-' +- fx~ z .x~* ,.&c.
^A-"~' -j- ax~ z fax" 3 .

Make_y = 2* 1 , &c. then; 1 = x 1 , 6cc. Therefore x~ z y = i ,
ficc. conlequently y = * -f- i -+- 3, &c. = * 4, 6cc. and therefore
j = * -+- 4.x, &c. Then x~ z y = * 4-v~", &c. and confe-
quently y = * * -f- o, 6cc. and therefore afluming any conftant
quantity a, it may be y = * * -+- a, &c. Then #-*_y = * *
^A,-" 1 , &c. and therefore , j = * * * -+- ax~* +- x~ z , &c. and
y=. * * * fix' 1 x" 1 , &c. And fo on. Here if we make

a = o, 'tis ^ = * 1 + 4* * ; H- ^ ^ 3 , &c.

51, 52. The Equation of this Example is y=. ^xy* -\-y, which
we fliall refolve by our ufual Method, without any other prepara-

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