Isaac Newton.

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

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19. So propofing the Equation yy = xy -+- xxxx ; I fuppofe x
to be the Correlate Quantity, and the Equation being accordingly

reduced, we mall have - = i -f- x 1 .v 4 -f- 2X & , &c. Now I mul-



tiply the Value of into x, and there arifes .v-f-AT 3 ' x f -{- 2X\

&c. which Terms I divide feverally by their number cf Dimenfions,
and the Refult x +- fv' fv'-f-fv 1 , &c. I put =y. And by

this



30 77je Method ^/"FLUXIONS,

this Equation will be defined the Relation between x and y, as was
required.

20. Let the Equation be - = a - -4- -f- ' 3 '* 3 &c. there

x 4 6-}<z 5i2*

will arife y = ax y -+- ~ j- -^ ' & c . for determining the

' y ZM OJ.oi.t~ o

Relation between A; and y.

21. And thus the Equation = _i_ -, , x* -t- #*,

v-J *.! I I

gives y = ^ -f- ^ . + 2 ^ |. x *+ ** . For multiply the
Value of - into A;, and it becomes - -f. ax^ - . x* -*- v*

*; Jf^ X X ,

or A:- 1 x' 1 -\- ax* x^-i-x^, which Terms being divided by
the number of Dimenfions, the Value of y will arife as be-
fore.

22. After the fame manner the Equation -. =5-7=== 4- -^ -+-

\/ f S7- 1. A



\- cy, gives A- = ^_ -}- H- - v/^)' 3 -i- cy~> . For the
Value of - being multiply'd by j, there arifes ~ -^ * _j_



-{-n'3 or 2^^-y* -h -~i ;' 3 + v/^ + c %y*. And thence
-the Value of x refults, by dividing by the number of the Dimen-
lions of each Term.

23. And fo =? =z\ gives y = $z*. And -1 =- 4 , gives r=
, ~ * 7

3f^L 3 . But the Equation ^ = ; , gives 7 = f . For f multiply'd

into A: makes a, which being divided by the number of Dimen-
fions, which is o, there arifes ~ , an infinite Quantity for the Value



_

24. Wherefore, whenever a like Term mail occur in the Value

of -. , whofe Denominator involves the Correlate Quantity of one

Dimenfion only ; inftead of the Correlate Quantity, fubftitute the
Sum or the Difference between the fame and fome other given
Quantity to be affumed at pleafure. For there will be the fame
Relation of Flowing, of the Fluents in the Equation fo. produced,
as of the Equation at firft propofed j and the infinite Relate Quan-

tity



and INFINITE SERIES. 31

tity by this means will be diminifh'd by an infinite part of itfelf,
and will become finite, but yet confifting of Terms infinite in
number.

25. Therefore the Equation 4 = - being propofed, if for x I
write ^4- x, affuming the Quantity b at pleafure, there will arife

v 11 T^ /* v fl a ^ ax^ ax^ c At

- = , : and by Divifion 4 = T rr 4- 77 -rr & c - And

u-^r~X * v O & b +

now the Rule aforegoing will give_}'= j - ^ 4- 3 ~p ~j^ &c. for
the Relation between x and y.

26. So if you have the Equation - = - 4-3 xx; becaufe

X X

of the Term ~ x -> if you write i -f- x for x, there will arife 4
. _f (_ 2 2X xx. Then reducing the Term ~-^ into an in-
finite Series 4-2 2x4- 2x l 2Ar 3 4- 2x% &c. you will have 4 ,

X

^ 4* _{_ x* 2x 3 4- 2x 4 , &c. And then according to the Rule
y = 4.x ax 1 4- fx 3 |x 4 4- ^x s , 6cc. for the Relation of x

and y.

27. And thus if the Equation -.-=x'^-i-x- 1 AT* were pro-



pofed j becaufe I here obferve the Term x l (or ~j to be found,
I tranfmute x, by fubftituting I x for it, and there arifes 4
. _' _ L __ - - v/ 1 A;". Now the Term - l x produces
i _{_ x _|_ x 1 4- x 3 , &c. and the Term \/i x is equivalent to

j, .i# 4-x 1 V^S an( ^ therefore or i _ v _ JL;( . a ^ is

the fame as i 4- -i-x 4- 4-x 1 4- |-x 3 , &c. So that when thefe Values
are fubftituted, I fhall have 4 = i ~f- 2x 4- 4x i 4-4-^-x 3 ,6cc. And

X

then by the Rule y =. x 4- x 1 4- 4-x* 4- ri* 4 , &c - An< i ^ oi
others.

28. Alfo in other Cafes the Equation may fometimes be con-
veniently reduced, by fuch a Tranfmutation of the flowing Quantity.

As if this Equation were propofed 4 = -^ ^^. c ^_ xi inflead



52 ^ Method of FLUXIONS,

O i/

of .v I write c AT, and then I mall have 4= ^ or 7 5 ~i>

and then by the Rule y = - J ^ -f,. L. But the ufe of fuch Tranf-
mutations will appear more plainly in what follows.

SOLUTION OF CASE II.

29". PREPARATION. And fo much for Equations that involve
only one Fluent. But when each of them are found in the
Equation, fiift it muft be reduced to the Form prefcribed, by
making, that on one fide may be had the Ratio of the Fluxions,
equal to an aggregate of fimple Terms on the other fide.

30. And befides, if in the Equations fo reduced there be any
Fractions denominated by the flowing Quantity, they muft be freed
from thofe Denominators, by the above-mentioned Tranfmutation
of the flowing Quantity.

31. So the Equation yax xxy aax = o being propofed, or

i_l _{_ f . becaufe of the Term -, I afiume b at pleafure, and

x a x *

for x I either write b -+- x, or b x, or x - b. As if I fhould
write b -+- x, it will become 4 = - -f- rrr. . And then the Term



being converted byDivifion into an infinite Series, we mall have

-1-1 , - - < , &C.

72. And after the fame manner the Equation = 37 2x +

J X

X 2v

- .. being propofed; if, by reafon of the Terms - and^.,

I write i y for y y and i x for x, there will arife =

X

_ o V -4- 2 x -f- ^- = -^ -4- 2 - v ~. 2 r . But the Term '- ^ by

3/ 1 y I ZX -\- X* 1 y J

infinite Divjfion gives i x -+-y xy -f-_y a xy* -J-_y 3 xy* t &c.
and the Term - t _^ 2 ~+ xx by a like Divifion gives 2_y 2 -i- ^xy
^ x _f- 6 x *-y . 6x a 4- S* 3 ^ 8x 5 + iox*y IOAT*, &c. There-
fore r-= 3^-i- 3^J -f->' a ' xy* -{- y3 ^y 5 , &c. -i- 6^^ 6x*

X

33-



and INFINITE SERIES. 33

33. RULE. The Equation being thus prepared, when need re-
quires, difpofe the Terms according to the Dimenfions of the flow-
ing Quantities, by fetting down fir ft thofe that are not affected by
the Relate Quantity, then thofe that are affected by its lead Dimen-
fion, and fo on. In like manner alfo diipofe the Terms in each of
thefe Clafies according to the Dimenfions of the other Correlate
Quantity, and thofe in the firft Clafs, (or fuch as are not affected
by the Relate Quantity,) write in a collateral order, proceeding to-
wards the right hand, and the reft in a defcending Series in the left-
hand Column, as the following Diagrams indicate. The work be-
ing thus prepared, multiply the firft or the loweft of the Terms in
the firft Clafs by the Correlate Quantity, and divide by the number
of Dimenfions, and put this in the Quote for the initial Term of
the Value of the Relate Quantity. Then fubftitute this into the
Terms of the Equation that are difpofed in the left-hand Column,
inftead of the Relate Quantity, and from the next loweft Terms
you will obtain the fecond Term of the Quote, after the fame man-
ner as you obtain'd the firft. And by repeating the Operation you
may continue the Quote as far as you pleafe. But this will appear
plainer by an Example or two.

34. EXAMP. i. Let the Equation 4 = i ^x-\-y-\- x*-{-.vy

be propofed, whofe Terms i T.V -+- A' 1 , which are not affected
by the Relate Quantity _v, you fee difpos'd collaterally in the up-





-h I T,X -\- XX


+'*,


* -+- A' X,Y-f-l.,V 3 ^.x-4_|__'_,v


r ,&c.

J_ ^ V

' "5""^"


s ,&c


The Sum


I ' 2.V " - I-"- &X * V ^ - 1 * v 4 i T ^_ \s


, &c.




y


A A -X -4*I - >4 + ^,__ Vx 6^ c .



permoft Row, and the reft ' y -and .vy in the left-hand Column. And
rirft I multiply the initial Term i into the Correlate Quantity .v,
.ind it makes x, which being divided by the number of Dimen-
fions i, I place it in the Quote under-written. Then fubftkuting
rhis Term inftead of y in the marginal Terms -f- y and -f- .vy, I
have -\-x and -+- xx, which I write over againft them to the right
hand. Then from the reft I take the loweft Terms ?.v and - x,
whofe aggregate zx multiply'd into x becomes 2.v.v, and

F being



3-4



The Method of FLUXIONS,



beino; divid'-d by the number of Dimenfions 2, gives xx for the
fecund Term of the Value of y in the Quote. Then this Term
being likewifc afiumed to compleat the Value of the Marginals -{-y
and -+- xv, there will arife alfo xx and x 5 , to be added to
the Terms -j-x and -{-xx that were before inferted. Which being
done, I again a flume the next loweil Terms -f-xx, xx, and -{-xx,
which I collect into one Sum xx, and thence I derive (as before)
the third Term -|-.ix ; , to be put in the Value of y. Again, taking
this Term -i-x 3 into the Values of the marginal Terms, from the
next loweft -f-y# 3 and x 3 added together, I obtain ^-x 4 for
the fourth Term of the Value of y. And fo on in infinitum.

35. Ex AMP. 2. In like manner if it were required to determine

the Relation of x and y in this Equation, y - -=. I -f- - -f- - v -f- r'-f-

< ^ a &* &*

- , &c. which Series is fuppofed to proceed ad infinitum ; I put I

in the beginning, and the other Terms in the left-hand Column,
and then purfue the work according to the following Diagram.





-hi




A" A* *3 .X 4


-.j


+ ~




h , , &c.


XV


A" a v 3 A 4


A *


4-


a 1 2^3 2^4


h z~ . &C -


X s " V


_1_ ' v3 i A ' 4


, . 5


4- ~




h , &c.


-4- ~


* * * * -+- -


h S ' &c -


4-*-?


* * * * * -


h-J , &c.


a*






Sum


.V 3** 2\= CAT4

T _l_ *_ 1 . 1 1

* I i ^ r i "^~"


3.V 5 c

h 4y , &c.




a ^a i a= z.;4




y ==


* + T a -+- ili + + ^ -


^ 6 o

h j , &c.



36. As I here propofed to extradl: the Value of y as far as fix
Dimenfions of x only ; for that reafon I omit all the Terms in
the Operation which I forefee will contribute nothing to my pur-
pofe, as is intimated by the Mark, &c. which I have fubjoin'd to
the Series that are cut off.

3 37-



and INFINITE SERIES. 35

37. EXAMP. 3. In like manner if this Equation were propofed



= 3,v -+- i*y -4-;* X j* -t-j 3 .vy 3 -4-;- A^

6..Y 1 -f- SA- J _V - 8.v 3 4- \oxy* IOA-*, &c. and it is intended to
extract the Value ot y as far as feven Dimensions of x. I place the
Terms in order, according to the following Diagram, and I work
as before, only with this exception, that iince in the left-hand Co-
lumn y is not only of one, but alfo of two and three Dimensions;
(or of more than three, if I intended to produce the Value of y
beyond the degree of x~* ,) I fubjoin the fecond and third Powers of the
Value of y, fo far gradually produced, that when they are fubftitu-
ted by degrees to the right-hand, in the Values of the Marginals





_ 3 . v _ 6X> 8* 3 IO.V^ I2A- M ,&CC.


+ 3*7


9 v ,

2"


6x*


b zo ' "


-+- 6x*y


* * *


gx*


I2.V ^V ,&C.


-f- 8*7


* * *


*


I2AT* l6x 6 ,fxc.


-f- IOA:^


* * *


#


* ^[J^ 6 j&C-


&c.








+-;*


* * #


^|*4


-f- 6x s -{-~^ 7 x 6 ,&;c.


xy*


* * *


*


4 * '


&C.








H-.v ;


* * *


*


* ~ - x s ,6cc.


Sum


3 A- 6x* ^f.v


3 9'

4


-^-'v' -Z.v-6 li-r-

^ * .X ,tXC.

h S '


3 2 S


qi


111 6 ^"


y= -A 1 2X> -*<


20


"16^ "77" r > C '


^ A ' 7
* " 4"^ 8


, &C.




y; x 6 , 6cc.







to the left, Terms may arife of fo many Dimenfions rs I obferve
to'be required for the following Operation. And by this Method

there arifes at length y= ^x 1 6.x 13 ^^+, &c. which is the

F 2 Equation



3 6 The Method of FLUXIONS,

Equation required. But whereas this Value is negative, it appears
that one of the Quantities x or y decreafes, while the other in-
creafes. And the fame thing is allb to be concluded, when one of
the Fluxions is affirmative, and the other negative.

38. EXAMP. 4. You may proceed in like manner to refolve the
Equation, when the Relate Quantity is affected with fractional Di-
menfions. As if it were propofed to extract the Value of x from

this Equation, - = iy ^y- -+- zyx* -J.v 1 -f- 77* -f- 2_y ; , in





H 5-7 * 4-y 1 -+ jy 1 + 2 >' 3


I


* * +)'* * 2_)' 3 -|-4}' T 2_y 4 , &c.
* * * * * * ~y 4 y &tc.


Sum


+ y # _ 3r _ f _ 7 / . +4/ 44-VS&C.


AT T == + 4_y y 1 -+- 2y* ' _)* , &c.
A;*= -V7 4 > ^ c -



which ,v in the Term a^'-x 11 (or zy^/x) is affected with the Frac-
tional Dimenlion -i- From the Value of x I derive by degrees the
Value of A?% (that is, by extracting its fquafe-Root,) as may be
obferved in the lower part of this Diagram ; that it may be in-
ferted and transfer'd gradually into the Value of the marginal Term
2yx' f . And fo at laft I fliall have the Equation x = .y l y* _|_
2_y^ -(- ^ T Vo^' f > & c - by which x is exprefs'd indefinitely in re-
ipect of y. And thus you may operate in any other cafe what-
foever.

39. I foid before, that thefe Solutions may be perform'd by an
infinite variety of ways. T'his may 'be done if you afiiime at
pleafure not only the initial quantity of the upper Series, but any
other given quantity for the firft Term of the Quote, and then you
may proceed as before. Thus in the firft of the preceding Exam-
ples, if you affume i for the firft Term of the Value of 7, and
fubftitute it for y in the marginal Terms -h_y and -t-xy, and pur-
fue the reft of the Operation as before, (of which I have here given a



and INFINITE SERIES.



37





-f- I 3x4- XV


4-*V


-4- i 4- 2x * 4- AT 3 4- .ix 4 , 6cc.

* -t- X 4- 2Ar l * 4- X 4 , &C.


Sum


4-2 * 4- 3** 4- A; 3 4-4-A" 4 , &c.


y - i -f- 2.v * 4- x"' -\- ix 4 4-^-A' 5 , 6cc.



Specimen,) another Value of y will arife, i -f- 2x-\- x* -h i* 4 , 6cc.
And thus another and another Value may be produced, by afTum-
ing 2, or 3, or any other number for its firfl Term. Or if you
make ufe of any Symbol, as a, to reprefent the firft Term inde-
finitely, by the fame method of Operation, (which I fhall here fet
down,") you will find y = a -+- x -+- ax xx -f- axx -+- ~x*+ax*,
&c. which being found, for a you may fubfHtute i, 2, o, 4-, or any
other Number, and thereby obtain the Relation between x and y
an infinite variety of ways.





4- i 3 x 4- A* AT


+y


_|_ fl _|_ x .v.V -


H yX 3 , &c.




4 #^" 4~ ^AT X -


f- -i^.v 3 , 6cc.


4-#y


* -f. tf .v 4- AT 1 -


- *s , &c.




-(- ^ZAT 1 -


f- ax* , &c.


Sum


4-1 2X 4- AT 1 -


AAr 5 , &C.




4-^4- 2^-4- 2x-


-f-l^x 3 , &c.


j = a 4- A; x 1


-h y-V 3 ^-.V 4 , &C.


4- ax 4- fl.v 1 -


f- j.tfJfJ + _ V ^ V 4 5 & C .



40. And it is to be obferved, that when the Quantity to be ex-
trailed is affected with a Fractional Dimenfion, (as you fee in the
fourth of the preceding Examples,) then it is convenient to take
Unity, or fome other proper Number, for its firft Term. And in-
deed this is neceflliry, when to obtain the Value of that fractional
Dimenfion, the Root cannot otherwife be extracted, becaufe oi
the negative Sign ; as alib when there are no Terms to be diJpofcd
in the firft or capital Clafs, from which that initial Term may be
deduced. 41.



38 tte Method of FLUXIONS,

41. And thus at laft I have compleated this moft troublefo'me
and of all others moft difficult Problem, when only two flowing
Quantities, together with their Fluxions, are comprehended in an
Equation. But befides this general Method, in which I have taken
in all the Difficulties, there are others which are generally fhorter, by
which the Work may often be eafed; to givefome Specimens of which,
ex abundantly perhaps will not be diiagreeable to the Reader.

42. I. If it happen that the Quantity to be refolved has in fome
places negative Dimenfions, it is not of ablblute necefllty that there-
fore the Equation mould be reduced to another form. For thus

the Equation y = - xx being propofed, where y is of one ne-
gative Dimenfion, I might indeed reduce it to another Form, as
by writing i -f- y for y ; but the Refolution will be more expe-
dite as you have it in the following Diagram.





#


* XX


I

y
Sum


i
i


V* - ! ^ V JK* ^CC


y


4- .V "'i-YAT -f- |-.V 3 , &C.

- x-t-^xx, 5cc.



43. Here affuming i for the initial Term of the Value of y., .
I extract the reft of the Terms as befoie, and in the mean time

I deduce from thence, by degrees, the Value of - by Divifion, and
infert it in the Value of the marginal Term.

44. II. Neither is it neceffary that the Dimenfions of the other
flowins Quantity fhould be always affirmative. For from the Equa-
tion y = 3 -\- zy '- , without the prefcribed Reduction of the

Term } ~ , there will arife_y = 3 A; xx -f- 2X J , &c.

4^. And from the Equation y = }'-+ - . ~ x > the Value

of y will be found y == ^, if the Operation be perform 'd after
the Manner of the following Specimen.



i

XX



and INFINITE SERIES,



3.9





I


.




*A:


.V






I


V


*


" .V


Sum


i


o




ATA:




y =




* X





46. Here we may obferve by the way, that among the infinite
manners by which any Equation may be refolved, it often happens
that there are fome, that terminate at a finite Value of the Quan-
tity to be extracted, as in the foregoing Example, And thefe are
not difficult to find, if fome Symbol be aflumed for the firft Term.
For when the Refolution is perform'd, then fome proper Value may
-be given to that Symbol, which may render the whole finite.

47. III. Again, if the Value of y is to be extracted from this

Equation y = ^. -+- i zx -f- xx y it may be done conveniently

enough, without any Reduction of the Term ~ , by fuppofing

(after the manner of Analyfts,) that to be given which is required.
Thus for the firit Term of the Value of y I put zcx, taking 2<? for
the numeral Coefficient which is yet unknown. And fubltituting
2.cx inftead of y, in the marginal Term, there ariies e, which I
write on the right-hand ; and the Sum i -f- e will give x -f- ex for
the fame firft Term of the Value of y t which I had firfi repre-
fented by the Term zcx. Therefore I make 2cx = x-}-ex, and
thence I deduce e =. i. So that the firfl Term zex of the Value
of y is 2.x. After the fame manner I make ufe of the fidlitious
Term 2/x* to reprefent the fecond Term of the Value of r, and
thence at laft I derive ^ for the Value of y, and therefore that fe-
cond Term is xx. And fo the fictitious Coefficient g in the
third Term will give T V, and b in the fourth Term will be o.
Wherefore iince there are no other Terms remaining, I conclude
the work is finiOi'd, and that the Value of y is exadtl-y zx x l
-if-^X', See the Operation in the following Diagram.



i



The Method ^FLUXIONS,





I ~2X +i XX


y






? 4~ /A* | - cfxx [ /yv'


Zx


6


Sum


4"~i ~~~ 2 A" 4~ XX


Hvpothetically r= zex-{- 2fx*-\- 2gx* 4- 2&c+

II II 1l II


Confequentially y= 4->v A* 4- ^x* 4- ^6^


Real Value j'= 2 A* l^ 1 4- ^-A-'



48. Much after the fame manner, if it were y = ^- ; fuppoie

y=.ex s , where e denotes the unknown Coefficient, and s the num-
ber of Dimeniions, which is alfo unknown. And ex' being fub-

ftituted for y, there will arife y =. - , and thence again 7 =

* . Compare thefe two Values of y, and you will find ^ = e,
and therefore s = , and e will be indefinite. Therefore afTuming

e at pleafure, you will have y = ex*.

49. IV. Sometimes alfo the Operation may be begun from the
higheft Dimenfion of the equable Quantity, and continually pro-
ceed to the lower Powers. As if this Equation were given, ^=:
2.1.1 _i_T_i_2;r -, and we would begin from the higheft

xx ~ XX ,. 3 * .

Term zx, by difpofing the capital Series in an order contraiy to the
foregoing ; there will arife at laft y = xx -f- 4.* - , &c. as may
be feen in the form of working here fet down.





4 '


+.i


* H- i 4-^ *


i i e
- -h > &C.

-v * ^A *r


Sum


i


rr + ^7* ' ^ cc>


_j> = A' 1 4- 4.v * ;


+ 1^ SIT > &c -



50.



and INFINITE SERIES, 41

50. And here it may be obferved by the way, that as the Opera-
tion proceeded, I might have inferted any given Quantity between

the Terms 4** and - , for the intermediate Term that is deficient,

and fo the Value of y might have been exhibited an infinite variety
of ways.

51. V. If there are befides any fractional Indices of the Dimen-
fions of the Relate Quantity, they may be reduced to Integers by
fuppofing that Quantity, which is affected by its fractional D-
menfion, to be equal to any third Fluent ; and then by ftibftitutii g
that Quantity, as alfo its Fluxion, ariling from that fictitious
Equation, inftead of the Relate Quantity and its Fluxion.

52. As if the Equation y= 3*7* -\- y were propofed, where the
Relate Quantity is affected with the fractional Index .1 of its Dimen-
fion; a Fluent z being afTumed at pleafure, fuppofe y^ = z, or
y = z'> ; the Relation of the Fluxions, by Prob. i. will be
y = 32Z 1 . Therefore fubftituting ^zz* for v, as alfo z* for y,
and z* for y$, there will arife yzz 1 = ^xz*- -+- z 3 , or z = x -\-^z,
where z performs the office of the Relate Quantity. But after the

Value of z is extracted, as z = x* -f- -f- ^ -J- -^- Q , &c. in-
ftead of z reftore y\ and you will have the defired Relation be-
tween x and v ; that is, y? = i.v 1 + -V^ 3 H- T-nr* 4 ; & c - an( ^ ^7
Cubing each fide, y =.^x 6 -\- T '_.v 7 -+- T Y T X S > ^ c -

53. In like manner if the Equation y = </^y -+- </xy were
given, or_y = 2^^ -J- xM ; I make z =)'^ or zz=y, and thence
by Prob. i. 2zz = y, and by confequence 2zz = 2z -f- x*z, or
z = i -+- {-x^. Therefore by the firft Cafe of this 'tis z = x -f-

-i-v 1 ", or y' 1 = Ar-f- -i.v 1 , then by fquaring each fide, v=y>; -+- -|Jf^
-i- -i-x 5 . But if you mould defire to have the Value of y exhibited
an infinite number of ways, make z =. c -f- x -f- -ytf , aiTuming any
initial Term c, and it will be ss, that is y, = c* -{- zcx + ^cx*
+ -v 1 -+- -i-x 1 * -t- ^v 3 . But perhaps I may feem too minute, in treat-
ing of fuch things as will but feldom come into practice.

SOLUTION OF CASE III.

54. The Refolution of the Problem will foon be difpatch'd, when
the Equation involves three or more Fluxions of Quantities. For

G between



42 ?$ Method of FLUXIONS,

between any two of thofe Quantities any Relation may be afiumed,
when it is not determined by the State of the Queftion, and the Re-
lation of their Fluxions may be found from thence ; fo that either
of them, together with its Fluxion, may be exterminated. For which
reafon if there are found the Fluxions of three Quantities, only one
Equation need to be affumedj two if there be four, and fo on j
that the Equation propos'd may finally be transform'd into another
Equation, in which only two Fluxions may be found. And then
this Equation being refolved as before, the Relations of the other
Quantities may be difcover'd.

55. Let the Equation propofed be zx z -f- yx = o ; that I
may obtain the Relation of the Quantities x, y, and z, whofe Fluxions
x, y, and z are contained in the Equation ; I form a Relation at
pleafure between any two of them, as x and y, fuppofing that x=y,
or 2y = a -+- z, or x=yy, &c. But fuppofe at prefent x=yy,
and thence x = 2yy. Therefore writing zyy for x, and yy for x,
the Equation propofed will be transform'd into this : q.yy z-^-yy*
= o. And thence the Relation between y and z will arife, 2yy-{-

^y= =.z. In which if x be written for yy, and x* for y~>, we mall
have 2X -f- ~x^ = z. So that among the infinite ways in which
x, y, and z, may be related to each other, one of them is here
found, which is reprefented by thefe Equations, .v =yy, 2y* +- y*
= z, and 2X -+- ^x* = z.

DEMONSTRATION.

56. And thus we have folved the Problem, but the Demonftra-
tion is ftill behind. And in fo great a variety of matters, that we
may not derive it fynthetically, and with too great perplexity, from
its genuine foundations, it may be fufficient to point it out thus in
fhort, by way of Analyfis. That is, when any Equation is propos'd,
after you have finifh'd the work, you may try whether from the
derived Equation you can return back to the Equation propos'd, by
Prob. I. And therefore, the Relation of the Quantities in the de-
rived Equation requires the Relation of the Fluxions in the propofed
Equation, and contrary-wife : which was to be fhewn.

57. So if the Equation propofed were y = x, the derived Equa-
tion will be y={x l ; and on the contrary, by Prob. i. we have
y xx, that is, y=.x, becaufe x is fuppofed Unity. And thus

from



and INFINITE SERIES. 4.3

from y = I 3* -+-y -f- xx -+- xy is derived _y = tf x* -f- Lx 1
^v+ -+- ^o x ! -4T' vS > &c - And thence by Prob. i. y = i 2x
^-x 1 %x> -+- ^-x* -V x! ) &c. Which two Values of y agree
with each other, as appears by fubftituting x xx+^x> -^x*
->-J-x s , <5cc. inftead of^ in the firft Value.

.,8. But in the Reduction of Equations I made ufe of an Opera-
tion, of which alfo it will be convenient to give fome account. And



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