Isaac Newton.

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

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plement of which is again the Root of an inferior Equation ; and fo on
for ever. Or retaining that Supplement, we may flop where we pleafe.
36. The Author's Diagram, or his Procefs of Refolution, is very
eafy to be underflood ; yet however it may be thus farther explain'd.
Having inferted the Terms of the given Equation in the left-hand
Column, (which therefore are equal to nothing, as are alfo all the
fubfequent Columns,) and having already found the firft Approxi-
mation to the Root to be a ; inflead of the Root y he fubflitutes its
equivalent a-\-p in the feveral Terms of the Equation, and writes
the Refult over-againfl them refpedtively, in the rightrhand Margin.
Thefe he collects and abbreviates, writing the Refult below, . in the
left-hand Column ; of which rejecting all the Terms of too high a
compofition, he retains only the two loweft Terms ^.a i p-\-a i x=.o^
which give p = x for the fecond Term of the Root. Then
afluming/> = -%x-}-q, he fubflitutes this in the defcending Terms
to the left-hand, and. writes the Refult in the Column to the right-
hand. Thefe he collects and abbreviates, writing the Refult below
in the left- hand- Column. Of which rejecting again all the higher
Terms, he retains only, the two loweft ^a*q T I T -cx i = o i which

give a for the third Term of the Root. And fo on.

Or in imitation of a former Procefs, (which may be feen-, ,pag;
165.) the Refolution of this, and all fuch like Equations, may be
thus perform'd.
i)3_|_tf^y= 2fl'= (if y=-a- i rp} a* + '$a*'p-\-T > ap !L -}-p* } Or collecting

-+.axy-h A? 3 + a * +ay > andexpung-

J ing,



r collecting and expunging,



I



X

By which Procefs the Root will be found _y = <z 7* 4- ^, &c.

Or



and INFINITE SERIES. 211

Or in imitation of the Method before taught, (pag. 178, &c.) we
may thus refolve the firft Supplemental Equation of this Example ;
w>. W-p -f- axp -f- 3^/ >a -4-/ 3 = a'-x -+ x* -, where the Terms
muft be difpos'd in the following manner. But to avoid a great deal
of unneceffary prolixity, it may be here obferved, that y = a, &c.
briefly denotes, that a is the firft Term of the Series, to be derived
for the Value of y. Alfo^=* f#, &c. infinuates, that fx

is the fecond Term of the fame Series y. Alfo y = * * -f-

644

&c. infinuates, that -4- r is the third Term of the Series y, with-

1 643 **

out any regard to the other Terms. And fo for all the fucceeding
Terms ; and the like is to be underftood of all other Series what-
ever.

4*'/l == a*x * +*

^ , &c.

40963

13 1x4 c

7T7T &c -




To explain this Procefs, it may be obferved, that here a*x is
made the firft Term of the Series, into which ^a l p is to be re-
folved ; or 4 t .a*p = a*x, &c. and therefore p = x, &c. which
is fet down below, Then is -f- axp = ^ax 1 , &c. and (by fquaring)
_f-3^) a = -t-Tz-ax l , &c. each of which are let down in their pro-
per Places. Thefe Terms being collecled, will make -V^S
which with a contrary Sign muft be fet down for the fecond Term

of ^a*p ; or 4d a /> = * + -r'^ax l } &c. and therefore p = * -f- -?- >

&c. Then axp=.*-^-^ , &c. and (by fquaring) 3<?/> a = * il. >
&c. and (by cubing) /> 5 = W^" 3 ' ^ c - T^efe being collected
will make ^, to be wrote down with a contrary Sign; and
this, together with A: 3 , one of the Terms of the given Equation,

will make a*p = * * -f- - 'x* } &c. and therefore />= * * -f- l -^- ?

i 1*4 s '*" a

&c. Then axp= * * -J- ~~ , &c. and (by fquaring) 3^/1* = * *

E e 2



212 Ih* Method of FLUXIONSJ

_, 1&22 & c . and (by cubing) * = * -f- -1^1 , &c. all which

4096.1 ' N ] ft! 1024* '

being collected with a contrary Sign, will make 4tf 1 /> = ***-f-
59i_* , &c. and therefore /=*** -f- ^ ,' &c. And by the

40961* 163841 '



fame Method we may continue the Extraction as far as we pleafe.

The Rationale of this Procefs has been already deliver'd, but as
it will be of frequent ufe, I fhaM here mention it again, in feme-
what a .different manner. The Terms of the Equation being duly
order'd, fo as that the Terms involving the Root, (which are to be
refolved into their refpecttve Series,) being all in a Column on one
fide, and t,he known Terms on the other fide ; any adventitious
Terms may be introduced, fuch as will be neceffary for forming the
feveral Series, provided they are made mutually to deftroy one an-
other, that the integrity of the Equation may be thereby preferved.
Thefe adventitious Terms will be fupply'd by a kind of Circulation, ,
which w^ill make the work eafy and pleafant enough ; and the ne-
ceffary Terms of the fimple Powers or Roots, of fuch Series as com-
pofe the Equation, muft be derived one by one, by any of the
foregoing Theorems. , - -

Or if we are willing to avoid too many, and to0 high Powers -
in thefe Extraction's, we may proceed' in the following manner.
The Example mall be the fame Supplemental Equation as before, ,
which may be reduced to this form, 4a* -f- ax -+- ^ap -4- pp -x.p =s

a*x * 4-# 3 j of which the Refolution may be thus : ,

" ' .'

-'! ' ' 4rf a H- ax



__, 3 . . .- $ - .

5 i 2a

-4-/ a - h TV** 77^ , &c:

X* I 3 I*S



6 4 5i2i 16384^3^



The Terms 4^* -f- ax-\- ^p-^-fp I call the aggregate Factor, of
which I place the known part or parts 4<2* -{- ax .above, and the
unknown, parts ^ap -f- pp in a Column to the left-hand, fa as that
their refpeclive Series, as they come to be known, may be placed
regularly over-againft them. Under thefe a Line is drawn, to receive

2, the



and INFINITE SERIES. 2*3

the aggregate Series beneath it, which is -form'd by the Terms of the
aggregate Factor, as they become known. Under this aggregate Se-
ries comes the fimple Factor />, or the fymbol of the Root to be
extracted, as its Terms become known alfo. Laftly, under all are
the known Terms of the Equation in their proper places. Now as
thefe laft Terms (becaufe of the Equation) are equivalent to the Pro-
duct of the two Species above them ; from this confideration the
Terms of the Series p are gradually derived, as follows.

Firft, the initial Term 4^ (of the aggregate Series) is brought
down into its place, as having no other Term to be collected with
it. Then becaufe this Term, multiply 'd by the firft Term of />,
fuppofe q, is equal to the firft Term of the Product, that is, ^.a'-q
= a*x, it will be q = ~x, cr p = -L.v, &c. to be put down
in its place. Thence we (hall have T > ap-=. %ax, &c. which to-
gether with -}-ax above, will make -^^'ax for the fecond Term ;
of the aggregate Series. Now if we fuppofe r to reprefent the le -
cond Term of p, and to be wrote in its place accordingly ; by crofs-
multiplication we lhall have ^.a^r -'y-ax^ =^ o, becaufe the fecond

_v^

Term of the Product is abfent, or=ro. Therefore r-=. , which

64*'

may now be fet down in its place. And hence yap = * -f- -^x 3 -,
&c. and p* = ^x*, &c. which being collected will make ^~x l ,
for the third Term of the aggregate Factor. Now if we fuppofe
s to reprefent the third Term of p, then by crofs-multiplication, (or
by our Theorem for Multiplication of infinite Series,) q.a 1 ; 4-

^ = * 3 ; (for x 3 is the third Term of the Product.) There-
256 256



fore s = l - , to be fet down in its place. Then -lap = * * -4-
512^*

&c. and i 1 = * A- , &c. which together will make

zea



5 I 2a

_j_ ^2l 3 for the fourth Term of the aggregate Series. Then putting
/.to reprefent the fourth Term of p, by multiplication we fliall have

^ = o, whence / = -^L to be

'



2048^ 4096*

fet down in its place. If we would proceed any farther in the Ex-
traction, we mufl find in like manner the fourth Term of the Se-
ries 3/, and the third Term of p*-, in order to find the fifth Term
of the aggregate Series. And thus we may eafily and furely carry
on the Root to what degree of accuracy we pleafe, without any
danger of computing any fuperfiuous Terms ; which will be no mean
advantage of thefe Methods.

Or



Method of FLUXIONS,

Or we may proceed in the following manner, by which we fliati
avoid the trouble of railing any fubfidiary Powers at all. The Sup-
plemental Equation of the fame Example, ^cfp +- axp -f- ^ap* -f-
p= = a*x-{-x*, (and all others in imitation of this,) may be

reduced to this form, /\.a*- -+- ax-+- ^a -{- p x/> x/> = a*x + x>,
which may be thus refolved.

4#* -f- ax







The Terms being difpofed as in this Paradigm, bring down 4^*
for the firft Term of the aggregate Series, as it may ftill be call'd,
and fuppofe q to reprefent the firft Term of the Series p. Then will
4^*5' = a*x, or^= ~x, which is to be wrote every where
for the firft Term of p. Multiply +- 3*2 by x for the firft Term
of 3#-f-/>x/>, with which product ax collect the Term above,
or -+- ax ; the Refult ~#x will be the fecond Term of the aggregate
Series. Then let r reprefent the fecond Term of />, and we fhal'l

have by Multiplication q.a'-r -r-s-^x 1 = o, or r = ^ , to be
wrote every where for the fecond Term of^>. Then as above, by crofs-
multiplication we fhall have 3^ x ~~ a -I- -rV-v 1 = ^V^ 1 ^ or tne third
Term of the aggregate Series. Again, fuppofing s to reprefent the
third Term of p, we ftiall have by Multiplication, (fee the Theorem,

for that purpofe,) A.a 1 s + , ', =x i . that is, s=^^ . to

' 256 256 5iz a '

be wrote every where for the third Term of p. And by the lame
way of Multiplication the fourth Term of the aggregate Series will be

found to be \-2L. } which will make the fourth Term -of p to be

And fo on.

Among all this variety of Methods for thefe Extractions, we
muft not omit to ftipply the Learner with one more, which is com-

mon



and INFINITE SERIES/

mon and obvious enough, but which fuppofes the form of the Se-
ries required to be already known, and only the Coefficients to be
unknown. This we may the better do here, becaufe we have al-
ready fhewn how to determine the form and number of fuch Se-
ries, in any cafe propofed. This Method confifts in the aflumption
of a general Series for the Root, fuch as may conveniently repre-
fent it, by the fubftitution of which in the given Equation, the ge-
neral Coefficients may be determined. Thus in the prefent Equa-
tion y= 4- axy 4- aay A' 5 2a 3 = o, having already found (pag.
204.) the form of the Root or Series to be y = A 4- Bx -+- Cx*, &c.
by the help of any of the Methods for Cubing an infinite Series,
we may eafily fubrtitute this Series inftead of y in this Equation,
which will then become



A 3 4~ 3 A l ijX 4~ ^n.D l x 1 4~ -D'AJ" 3 4~ 3 ^"*> *^c*
4^ 3A*C 4- 6ABC4- 36^
} 4- 6ABD



aA.x 4- aBx 1 - 4- aCx* -f- aDx*, &c.



o.



Now becaufe x is an indeterminate quantity, and muft continue'
fo to be, every Term of this Equation may be feparately put equal to-
nothing, by which the general Coefficients A,B, C, D, &c. will be de-
termined to congruous Values ; and by this means the Root^ will be
known. Thus, ( i.) A 3 4- a 1 A 2a~> = o, which will give A=/r,



as before. (2.) 3A a B -+- aA -+- a*-B = o, or B==



(4.) B 3 -4-6ABC-j-3A*D4-rfC-H a D i = o, orD'=^_>
(5.) 3 AO + 3B-C + 6ABD 4- 3 A*E 4- aD 4- ^E = o, or =

_^2_ . And. fo on, to determine F, G, H, &c. Then by fubfti-

163^4^^

tuting thefe Values of A, B, C, D, &c. in the aflumed Root, we

(hall have the former Series y =ax + ^4- '+ ;%^> &c.
Or laftly, we may conveniently enough refolve this Equation, or
any other of the fame kind, by applying it to the general Theorem,
ra CT . 1 90. for extracting the Roots of any affedted Equations in Num-
bers. For this Equation being reduced to this form ; i3 * 4- a 1 4-^x

2. x/



2l b The Method of FLUXIONS,

x 2rt 3 -+-A:' x.y c = o, we fliall have there #2 = 3. And inftead
of the firft, fecond, third, fourth, fifth,- &c. Coefficients of the Powers
of y in the Theorem, if we write 1,0, aa -f- ax, 2# 5 x ? , o,

.&c. refpectively ; and if we make the firft Approximation - = - >
or A= a and B = i ; we. fliall have 4 " , , A * for a nearer Approxi-

4 a -f- *

mation to the Root. Again, if we make A= 4^' -f- x*, and
B 4^ -f- ax, by Subftitution we fliall have the Fraction



.t5 -(- 48a*.v4 -f- i Zfi4 ,* -f- zqSjt .* * +1*9 ,.



nearer Approximation to the Root. And taking this Numerator
for A, and the Denominator for B, we fliall approach nearer ftill.
But this laft Approximation is fo near, that if we only take the firft
five Terms of the Numerator, and divide them by the firft five
Terms of the Denominator, (which, if rightly managed, will be no
troublefome Operation,) we fliall have the fame five Terms of the
Series, fo often found already.

And the Theorem will converge fo faft on this, and fuch like oc-
-cafions, that if we here take the firft Approximation A = a, (ma-

king B = i ,) we fliall have y = -^ ^ ** , &c. = a ~x, &cc.
.And if again we make this the fecond Approximation, or A a
t*, (making B = i,) we fliall have y =



4 a ax ~T -i * 4 5 1 z 4

if again we make this the third Approximation, or A=:a



_ &c. (making 'B== J,) we fliall have the Value of the

D ti& ^ * *-**

true Root to eight Terms at this Operation. For every new Oper-
ation will double the number of Terms., that were found true by the
laft Operation.

To proceed ftill with the fame Equation ; we have found before,
pag. 205, that we might likewife have a defcending Series in this
form, v = AA'H-B -j-Cx- 1 , &c. for the Root y, which we fliall
extract two or three ways, for the more abundant exemplification of
this Doctrine. It has been already found, that A= i, or that x
is the firft Approximation to the Root. Make therefore y =. x
and fubftitute this in the given Equation jy 3 -f- axy -f- any x>
*a= = o, which will then become ^p -f- axp -f- a?p -\- ^xf
-4- ax 1 -f- a^x 2 3 = o. This may be reduced to this form



-^ _t_ ax -+- a* -f- 3.v/> -{-/* x/> = ax"- a*x + 2a*, and may
be refolved as follows. OAT*



and INFINITE SERIES. 217

3A' 1 -f- aX -f- rt*



ax _ a * _+_ IE1 t Sec.

4_ p* . _ + ^ _+_ _; s &c>



3-v* * -f- t rfl +- ^ , &c.

/,.. _i_^ -U_ *""* _i- 64 " 4 c -

y ' 3 " 3-v ~~ 81^^ ~



The Terms of the aggregate Factor, as alib the known Terms of
the Equation, being difpofed as in the Paradigm, bring down ^x l
for lire firft Term of the aggregate Series ; and fuppofing q to repre-
lent the firft Term of the Series p, it will be 3^^ = ax*, or
q=- y, for the firft Term of p. Therefore ax will be the
firft Term of 3^ to be put down in its place. This will make the
fecond Term of the aggregate Series to be nothing ; fo that if ; re-
prefent the fecond Term of p, we fliall have by multiplication 3vV

= a 1 *;, or / = "_ for the fecond Term of p, to be put down

in its place. Then will a 1 be the fecond Term of $xp, as alfo
^d"~ will be the firft Term of/ 1 , to be fet down each in their places.
The Refult of this Column will be -^z 1 , which is to be made the
third Term of the aggregate Series. Then putting s for the third
Term of/, we mall have by Multiplication ^x^s -V rt3 == 2(l ' >
or s= $52- . And thus by the next Operation we fhall have / =

1 J



and fo on.

"Or if we would refolve this reildual Equation by one of the fore-
going; Methods, by which the railing of Powers was avoided, and
wherein the whole was performed by Multiplication alone ; we may

reduce it to this form, 3* 1 + fix -f- a 1 -f- 3* -j-/ x/> x/ = ^.v 1
d^X _j_ 2n* , the Refolution of which will be thus :



F f j.v-



2I 8 tte Method of FLUXIONS,



3** -f- ax -h a* *
- + 3* T* ~



a

fa* + , &c.






- - .

3*- 8ix* 243*3'



The Terms being difpos'd as in the Example, bring down 3*'* for
the firft Term of the aggregate Series, and fuppofing q to reprefent
the firft Term of the Series p, it will be yx^q = ax*, or q =
La. Put down -+- 3* in its proper place, and under it (as alfo after
it) put down the firft Term of/, or La, which being multiply'd,
and collected with -j- ax above, will make o for the fecond Term
of the aggregate Series. If the fecond Term of p is now reprefented

by r, we fhall have ix^r * = a'-x, or r = , to be put

3^*
down in its feveral places. Then by multiplying and collecting we

mail have -f- a* for the third Term of the aggregate Series. And
putting s for the third Term of p, we fhall have by Multiplication

3Ar*j T ' T rf3 =2d 3 , or j= |^ . And fo on as far as we pleafe.

Laftly, inftead of the Supplemental Equation, we may refolve the
given Equation itfelf in the following manner :



*

28*4
ax 1 %a*x -f- \a* - , Sec.

- - - f- ax 1 La*x La* +- , &c.






y= x 'a



243^5



Here becaufe it is y~> =x*, &c. it will be y = x, &c. and therefore
_t_ xy =-f- A -I , &c. which muft be fet down in its place. Then
it muft be wrote again with a contrary fign, that it may be y= == *
rfx*, &c. and therefore (extracting the cube-root,) /= * a
&c. Then -+- a*y = 4- a*x, &c. and + axy = * , j-^^, &c.

which



and INFINITE SERIES. 219

which being collected with a contrary fign, will makers = * *
JLa*x, &c. and (by Extraction) y = * * , &c. Hence -f- a l y

= # frt 3 , &c. and -f- ^v>'= * * a*, &c. which being col-
lected with a contrary fign, and united with -f- 20 J above, will

make y"' = * * * f^ 5 , &c. whence (by Extraction) y = * * * 4-^ >

&c. Then -+- a 7 = * * j*, &c. and -f- axy = * * * + ^7'
&c. which being collected with a contrary fign, will make y* =
* * * * 177 ' &c ' and l ^ en (by Extraction)^' ==* * * * +



&c. And fo on.

37, 38. I think I need not trouble the Learner, or myfelf, with
giving any particular Explication (or Application) of the Author's
Rules, for continuing the Quote only to fuch a certain period as {hall
be before determined, and for preventing the computation of fuper-
fluous Terms ; becaufe mod of the Methods of Analyfis here deli-
ver'd require no Rules at all, nor is there the leaft danger of making
any unneceflary Computations.

39. When we are to find the Root y of fuch an Equation as

this, y t)' 1 -+- fj 3 t v4 + t>"> &c - = *> tllis is ufually call'd
the Reverfion of a Series. For as here the Aggregate z is exprefs'd by
the Powers of y; fo when the Series is reverted, the Aggregate y
will be exprefs'd by the Powers of z. This Equation, as now it
(lands, fuppofes z (or the Aggregate of the Series) to be unknown,
and that we are to approximate to it indefinitely, by means of the
known Number y and its Powers. Or otherwife ; the unknown
Number z is equivalent to an infinite Series of decreafing Terms,
exprefs'd by an Arithmetical Scale, of which the known Number y
is the Root. This Root therefore muft be fuppofed to be lefs than
Unity, that the Series may duly converge. And thence it will fol-
low, that z, alfo will be much lefs than Unity. This is ufually cal-
led a Logarithmick Series, becaufe in certain circumftanp.es it ex-
preffes the Relation between the Logarithms and their Numbers, as
will appear hereafter. If we look upon z, as known, and therefore
y as unknown, the Series mull be reverted; or the Value of y muft
be exprefs'd by a Series of Terms compos'd of the known Num-
ber z and its Powers. The Author's Method for reverting this Se-
ries will be very obvious from the confideration of his Diagram ;
and we mall meet with another Method hereafter, in another part of
his Works. It will be fuffiqient therefore in this place, to perform it
after the manner of fome of the foregoing Extractions.

F f 2 y



Method of FLUXIONS,

y 1 = a + |~ a -+- f:i 3 + T V- 4 4- T5o-3 } > &c -

f V- > * h f * 5 -4- 'f^ 4 -H AS',' &c.

f S 4 _. fa', &c.

M L. AX* &C.

A./ -s J *

Sec.

In this Paradigm the unknown parts of the Equation are fet down
in a defcending order to the left-hand, and the known Number z is
fet down over-againft y to the right-hand. Then is y = z, Sec.
and therefore fj* = fa 1 , Sec. which is to be fet down in its
place, and alfo with a contrary fign, fo that _}'= * -f- f % & c -
And therefore (fquaring) f^ 1 = * f 2', Sec. and (cubing)
-h fy 3 =4- fa 3 , Sec. which Terms collected with a contrary fign,
make y= * * -f- -.^s, Sec. And therefore (fquaring) f_y* =
* * rV 24 & c - an d (cubing) -4- f_)' 3 = * -|- fa 4 , Sec. and f/ 4
=: f.?. 4 , Sec. which Terms collected with a contrary fign, make
y = ***-{- -j 1 -?- 54 ' ^ c - Therefore y_y a = * # * f.s f , &c.

H- f^ 5 = -{- f^% &c. which Terms collected with a contrary fign,
make y= #- ***-{- -4-a s , Sec. And fo of the reft.

40. Thus if we were to revert the Series y -f- f/ 3 -f- ^V>' 5 + TT-T^
-f- T .fy T y' -h TTTS->''S ^ c - = ^, (where the Aggregate of the Se-
ries, or the unknown Number a, will reprefent the Arch of a Circle,
whole Radius is i, if its right Sine is reprefented by the known
Number y,) or if we were to find the value of r, confider'd as un-
known, to be exprefsd by the Powers of a,, now confider'd as known ;
we may proceed thus :

Lo*3 ,^1^ f ]_ o* 5 ____'__ *?" 1 -.{ ^9 o^C

+ 3 5 i>>9 ATr*

T"T"3""a" 3 vVv

Sec. j

The Terms being difpofed as you fee here, we mall have jy==a,
Sec. and therefore (cubing) fj 5 = fa 3 , Sec. which makes y = *
fs 3 , Sec. fo that (cubing) we lhall have + f_>' 3 = * -rV^'j
Sec. and alfo -^y 1 =- 5 ? 5 -a !r , Sec. and collecting with a contrary fign,

y



and INFINITE SERIES, 221



r==* * -+-TT.T-'. &c - Hence \<-> = * * T ' T V 7 > & c. and ^y
' * T V~ 7 , &c- and TTT.v 7 = TTT~ 7 > &< - . and collecting with a
contrary tign, v = * * * WT^"' &c< Anct lo on '

It" we fhould defire to perform this Extraction by another of the
foregoing Methods, that is, by fuppoiing; the Equation to be reduced

to this- form i -+- j-_v* 4- -rV 4 +- TTT.' 6 + TTTT^'^ &<' x ;==;, it
may be fufHcient to let down the Praxis, as here follows.



I


* * * *


-f- f y*


h i" 5 -' 1 V"-' 4 4~ TTT^ "~~ TT'po-' 2 > ^ c -


1 3 V 4


1 I A i t ft &




4 ^ _j_ s^ "s - 8 ' &c


FT^s


' 11 " 3 5*. ,9* ^


1 ' . T * .*


1 . i 5 l >




41. The afFedted Cubick Equation, which the Author here affumes
to be folved, has infinite Series for the Coefficients of the Powers of
y ; and therefore its Terms being difpoled (as is taught before) accor-
ding to a double Arithmetical Scale, the Roots of each of which are
V and z,, it will ftand as is reprefented here below. Or taking As"
for the fir ft Approximation to the Root y, and lubftituting it in the
firft Table, it will appear as is here let down in the fecond Table.

. * S 1 . * *

A -. ; ,-'"-H- 4. Ai m + z +



J2-V. C-V. 45V. &f. J

Now the only cafe of external Terms, to be difcover'd by apply-
ing the Ruler, will give the Equation A 3 * m + 8 = o, whence
j;-|-2=o, or w= .1, and the Coefficient A = 2. The
next Number or Index, to which the Ruler in its parallel motion
will apply itfelf, will be 2m H- 2, or .1 ; the next will be m -f- 2.,
or ; and fo on. Which afcending Arithmetical Progreflion o, |,
i, 6cc. will have ~ for its common difference. Therefore y_ A.g~
-f- B +Cs^+ D^J -{- E^, &c. will be the form of the Root in this
Equation. It may be refolved by any of the foregoing Methods,

bat



The Method of FLUXIONS.



222

but perhaps moft readily by fubftituting the Value of y now found
in the given Equation, and thence determining the general Coeffi-
cients as before. By which the Root will be found to be _)' =

or J - I- Z^ I 3 '_ *;7 i * *_2- I 9 9 fy7 I 6 i i i ~ 3 f, _
Z.O 3 gt** TT T i ~ TTTT' 6 T^ T JTT^rT > **-*-.

42. To refolve this affected Quadratick Equation, in which one
of the Coefficients is an infinite Series ; if we fuppofe y =. Ax m , &c.
we (hall have (by Subftitution) the Equation as it ftands here below.

Then by applying the Ruler, we {hall have aAx m -+- 4 =o,

whence m = 4, and A = ~ . The next Index, that the Ruler

in its parallel motion will arrive at, is m -+- I, or 5; the next is
m-\-2, or 6; &c. fo that the common difference of the Progrel-
fion is i, and the Root may be reprefented by y = Ax* -{-Ex* -f-
Cx 6 , &c. which may be extracted as here follows.



x"


*
aAx


&e. "


*


. m-l-i

A*


*




A M-J-S




#


~ X

a


*




A_ +?


*


*


^^~ j_X






_ A VH


AT4


*




?



-ay

xy
x*



X4

-7^

6T*.

-fy



4-2



A

,77 7 . &c -



Here becaufe it is ay = . . &c. it will be y = fl

4 J 4^5 >

Therefore xy = ^ , &c, which wrote with a contrary Sign



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