FLU
(64)
FLU
The latter is directly oppofite to the former ; and is a ... Sequel of it. Both of them are adopted into the new Geo- scion of x v y z, is x v y z -|- x v y z -~ x v y raetry, and make reigning Methods therein. .
The firft defcends from finite, to infinite; the latter af- cends from infinitely fmall, to finite : The one decompounds a Magnitude, the other re-eftablifhes it*
2)irecJ Method of Fluxions.
All finite Magnitudes are here conceiv'd to be refolv'd into infinitely fmall ones 5 which are the Elements, Mo- ments, or Differences thereof.
The Art of finding thefe infinitely fmall Quantities, and of working on them, and difcovering other infinite Quanti- ties by their means, makes the 2)ire£t Method of Flu- xions.
What renders the Knowledge of infinitely fmall Quanti- ties of fuch infinite Ufe and Extent, is that they have Re- lations to each other, which the finite Magnitudes, where- of they are the Infinitefimals, have not.
Thus, e.gr.'ln a Curve of any kind whatever, the infi- nitely fmall Differences of the Ordinate and Abfcifle, have the Ratio to each other, not of the Ordinate, and Abfciffe; but of the Ordinate and Subtangent; and of confequence, the Abfcifle and Ordinate alone being known, give the Sub- tangent unknown; or which amounts to the fame, the Tan- gent it felf. See the inverfe Method of Fluxions.
The Method of Notation in Fluxions, introdue'd by the Inventor Sir /. Newton, is thus ,
The Fluxion of the variable, or flowing Quantity, to be uniformly augmented, as fuppofe the Abfcifle of a Curve, he denotes by 1, or Unite 5 and the other flowing Quantities he denotes by the Letters v x y z ; and their Fluxions by
the fame Letters, with Dots plac'd over themj thus v x
y z.
Apain, as the Fluxions themfelves are alfo variable Quan- tities, and are continually increafing, or decreafing ; he con- siders the Velocities with which they increafe, or decreafe, as the Fluxions of the former Fluxions, or fecond Fluxions-,
which are denoted with two Dots over them, thus, y x z.
After the fame manner one may confider the Augments,
and Diminutions of thefe, as their Fluxions alio; and thus
proceed to third, fourth, fifths &c. Fluxions, which will be
V x z &c.
x v y z ; and the Fluxion of a -1- x x by h—y (the com- mon Product being a b A- b x — y a — x y) will be b x — y a — x y ■
x y.
noted thus, y x z : y x z : y
Laftly, if the flowing Quantity be a Surd, as ^ -. a—b ;
he notes its Fluxion *J : a V .- If a Fraction — ' he
notes it, — : — : rf— y
The chief Scope and Bufinefs of Fluxions, is, from the flowing Quantity given, to find the Fluxion: For this, we /Kail Jay down one general Rule, as ftated by Dr. Wallis ; and afterwards apply and exemplify it in the feveral Cafes.
" Multiply each Term of the Equation feperatelyby the " feveral Indices of the Powers of all the flowing Quanti- t( ties contain'd in that Term : And in each, change one " Root or Letter of the Power into its proper Fluxion : " The Aggregate of all the Produces connected together " by their proper Signs, will be the Fluxion of theEqua- " tion defired."
The Application of this Rule will be contain'd in the following Cafes. — —
L In the General ; To exprefs the Fluxions of fimple variable Quantities, as already mentioned, you need only put the Letter, or Letters, which exprefs them, with a Dot over them: Thus, the Fluxion of x is * ; and the Fluxion
of_y is j 5 and the Fluxion of x-\ y~\-vA-z, is x-\-y-\~v-\-z
&c.
Note. For the Fluxion of Permanent Quantities, when any fuch are in the Equation, you muft imagine o, or a Cypher^ for luch Quantities can have no Fluxion, properly fpealung, becaufe they are without Motion, or invariable.
II. To find the Fluxions of the TroduBs of two or more •variable or flowing Quantities ;
Multiply the Fluxion of each fimplc Quantity by the Factors of the Produces ; or the Product of all the reft ; and connect the laft Produces by their proper Signs; the Sum or Aggregate is the Fluxion fought.
Thus,' the Fluxion of .1? y is x y A- x y ; and the Flu-
xion of x y z> is x y z *\- x y z 4- x y z ; and the Flu-
III. "To find the Fluxion of a Fra&ion ;
Multiply the Fluxion of the Numerator by the Denomi- nator, and after it place (with the Sign — ) the Fluxion of the Denominator to the Nominator, and divide the whole by the Square of the Denominator.
X
Thus, the Fluxion of — is x y—x y ;
y
For fuppofe — — z, then will x— j<s; which equal
Quantities will have equal Fluxions ; therefore .v ~ y z-\-
. x — x y z y , and x — z y =iz y $ and dividing all by y ~~Z "
x * -
= z — ( becaufe — = 3) yx -xy : Wherefore this
y ■ —
9 y
laft is the Fluxion of the Fraction —5 becaufe z being =
y
x : _ x__
, z will be equal to the Fluxion f y
a ~xa
And the Fluxion of— will bey; ; for the permanent Quantity a having no Fluxion, there can be no Product of the Fluxion of the Numerator into the Denominator, as there would have been, had a been x, s, or any other variable Quantity.
IV. 1q find the Fluxion of a 'Power;
Multiply the Power (firft brought one Degree lower) by the Index of that firft Power ; and the Frodud by the Fluxion of the Root.
Thus, the Fluxion of x x will be 2 * *, for x x ■=. x x #5
but the Fluxion of * x z = * * 4- x x — % x Xj gfc and
the Fluxion of x 3 will be 3 x x x. That of x* will be 8 x 7
x, &c. Or if m exprefs the Index of any Power, as fup-
tofe x"> its Fluxion will bew*"- 1 *? or m x x »— • ': For x m brought one Degree lower (m being a general In- dex) mufl be * m — " j then that x by m the Index, makes m x m—*. and this laft by the Fluxion of the Root, pro- duces m x * — * *. , ■
If the Power be produced from a Binomial,^, as lup-
pofe x x A- 2 x y -j- y y, its Fluxion will be a x x J^ 2 x x 4- z x y 4- 2 y y> by the Fourth and Second Rules. If the Exponent be Negative, as fuppofe x — » or x n,
its Fluxion will be — m **— th?-* *.
Or, if vou would do it by way of Fraction, ■
( for. the Square of x"> is as well x 2m as x ">*■ ) or according to
— m x Sir LNcwtou's Method, which is yet fhorter, ~ m +. l .
If the Power be imperfect, i. e> if its Exponent be a Fra- ction, as fuppofe »J : « m ; or in the other Notation x IT,
fuppofe * " n '= z : Then if you raife up each Member to the Power of n, it will ftand thus, x»» == a »■ the Fluxion of
which will be, by this general Rule, m x » — 1 x =r «
m x x >* — x
a „ _- 1 x.\ Wherefore z will be = - ^ m _ l ( by dividing
mxx m ~ l m m _~ 1 ' both Parties by n z »-i ») and — -— = * » -^5
L m :. x » »»**, by putting inftead of n z™-<\ its Va-
m
lue n x " » .
Hence
