Charles Hutton.

A course of mathematics ... For the use of academies, as well as private tuition online

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Ex^m. 4. To find the correct fluent of i = ■■ ?*f- ; sun-

a + x r

posing z and x to begin to flow together, or to be each
= at the same time.

Exam. 5. To find the correct fluents of £ =3 ** • sun-

a* + x* ' r
posjng z and x to begin together.

Art. 49.

Digitized by


t 333 J


Art. 49. In art 42, &c. is given a compendious table of
various forms of fluxions and fluents, the truth of which it inay
be proper here in the first place to prove.

50. As to most of those forms indeed, they will be easily-
proved, by only taking the fluxions of the forms of fluents,
in the last column, by means of the rules before given in
art. 30 of the direct method » by which they will be found to
produce the corresponding fluxions in the 2d column of the
table. Thus, the 1st and 2d forms of fluents will be proved
by the 1st of the said rules for fluxions : the 3d and 4th forma
of fluents by the 4th rule for fluxions ; the 5th and 6th forms,
by the 3d rule of fluxions: the 7th, 8th, 9th, 10th, 12th. 14th
forms, by the 6th rule of fluxions: the 17th form, by the 7th
Tule of fluxions : the 18th form, by the 8th rule of fluxions.
So that there remains only to prove the 1 lth, 13th, 15th, and
16th forms.

51. Now, as to the 16th form, that is proved by the 2d ex-
ample in art. 98, where it appears that x^(dx—x* ) is the
fluxion of the circular segment, whose diameter is rf, and
versed sine x. And the remaining three forms, viz, the 1 lth,
13th, and 15th, will be proved by means of the rectifications
of circular arcs, in art. 96.

52. Thus, for the 1 lth form, it appears by that art that the
fluxion of the circular arc z, whose radius is r and tangent f,

is z = j^rp Now put t = •r* B , or t* = **, and a = r» :
then is J =r jfiar 1 "" x> andr*+ ** = « + x\ and z » —^
= if?;* i ; hence ±^± _* e i ; ^ thc flucm .

a+x** <i+x» jpn an

2s 8 in 2

— = — x arc to radius */a and tang, x* or =. x are

an na w n*/a

to radius 1 and tang. </ — , which is the first form of the

fluent in n°. »•

53. And, for the latter form of the fluent in the same n° ;

because the coefficient of the former of these, viz, -2-, is
double of the coefficient of the latter, therefore the arc


in the latter case, must be double the arc in the former.
But the cosine of double an arc, to radius 1 and tangent f , is

Digitized by



l -^-i and Urns* ** « -i" by the former case, this substi-

1 — <* <* — ar* .

tuted for ** in the cosine - — , it becomes -rrrii the co-

«m as in die latter ease of the 1 1th form.

54. Again, for the first case of the flues* in the 13th
form* By art. 61, the fluxion of the circular arc 2, to radius r

and sine y, is * = ^L^yor =jg~*y to the radius i.

Nowputy = V— >o*y* =— i bencc* • (I - £*) =

y/ <1 _ ?J)= v 'Ixv'(a-a*),and; = ^ i Xi** 4 ""*;
then tbeie two Being substituted in the value <£ i, girt i

«* •(vnrr ? * 7ffA» n****** *• *** flyxkm

* ""* ris=-i, and therefore its fiueht i»|-*> «*« »

~ X arc to sine %/ ^ , as in the table of forms, for the first

h a

case of form xm.
5 5. And, as the coefficient ~ , in the latter case of the Said

form, is the half of — the coefficient in the former case,

therefore the arc in the latter case must be double of th* s#e
in the former. But, by trigonometry, the* versed sine of
double an arc, to sine y and radius l,ia2y»; aad, by the

former case, 2y» 3= ^ • therefore ^ x a* c to the versed si**

a " ,

— is the fluent, as in the 2d case of form xin.

56. Again, for the first case of fluent in the 15th form.
By art 61, the fluaion of die circular arc z, to radius r

and secant *,isi=-^ radius 1.

Now, put # = v— = — , or *« «= ~ ; hence •VC* - 0**
a Ja «

i w ^m i n 1 i 1 *"" 1

then these two being substituted in the value of % glre * of

■ ■ ■ «/>. . «* > =i^~X -^^^cohsequraUythe*^^

^*- l i 2 2 2

-7 - — r = — i, and theref. its fluent is — -z, that is


Digitized by



X n

X arc to secant.*/ ^-, as in the taplc of forms, for the first
case of form xv.

57. And, as the coefficient , in the latter case of the

said form, is the half of -^— the coefficient of tho former

case, therefore (he arc in the latter case mist be double the
arc in the former. But, by trigonometry, the cosine of the

double arc, to secant « and radius 1, is y -=•• i ; and, by the

3 2a %a~x* . - 1

former case, 1 = -~ — 1 ta — ; therefore — - x

«■ * n x n ntja

arc to cosine _iz£2 \ % the flucnt,as in the 2d case of form xv.
x n

Or, the same fluent will be x arc to cosine </ — , be-

«•« x»

cause the cosine of an arc, is the reciprocal of its secant.

58. It has been just above remarked, that several of the
tabular forms of fluents are easily shown to be true, by taking
the fluxions of those forms, and finding they come out the
same as the given fluxions; But they may also be deter*
mined in a more direct manner, by the transformation of the
given fluxions to another form. Thus, omitting the first
form, as too evident to need any explanation, the 2d form is
' z :== (a+ & y*^ 1 *"- 1 *, where the exponent (n — 1) of the
unknown quantity without the vinculum, is 1 less than (»)
that under the same. Here, putting y =*= the compound

quantity a + x* : then is y = h*"^ 1 *, and i => l s- ; hence,

by art. 36, z> — • £ — - *» "- — — as in the table.

59. By die above example it appears, that such form of
fluxion admits of a fluent in finite terms, when the index
(n -r 1) of the variable quantity (or) without the vinculum,
b less by 1 than *, the index ?f the same quantity under the
vinculum. But it will also be found, by a like process, that
the same thing takes place in sqtb forms as (a + x tl ) m x cn " 1 ^ i
where the exponent (en— 1) without the vinculum, is 1 less
than any multiple (c) of that (n) under the vinculum. And
further, that the fluent, in each case, will consist of as many
terms as are denoted by the integer number c; viz, of one
term when c = i,of two terms when e *» 2, of three terms
when c = S, and so on,

60. Thus, in the general form, i = (a + x n y*aF"*£,
putting as before, a -f x« ** y \ *hen is x n = y - c, and its


Digitized by VjOOQlC


fluxion nxP-ix as y . or *"-»i = *-, and *«»-»i or *«■-•

x*~ l x = — (y — a)*" 1 ^ ; also (a + x*y* = y m s these va-
lues being now substituted in the general form proposed,
give$ = — (y —ay-iy^y. Now, if the compound quan-
tity (y — oif-i be expanded by the binomial theorem, and
each term multiplied by y"»j, that fluxion becomes

x= I(y—V - f-=J«y"M^; + c -^-i . c -ll as y «^; _
fcc) ; then the fluent of every term, being taken by art. 36, it is

1 y"> + c C»l ay»U*-l C— 1 t"2 fljym+r-2

y* ,t _c~l a c—l.c-2 a* c— l.-2.<— 3 a»
"" n ^""(/-i'J + d-2 "2y* <*-3 *S.3y*

Ice), putting d = m + c, for the general form of the fluent ;
where, c being a whole number, the multipliers c- I, c-2,
r— 3, &c, will become equal to nothing, after the first c
terms, and therefore the series will tl:en terminate, and ex-
hibit the fluent in that number of terms; viz, there will be
only the first term when c =. 1, but the first two terms when
e = 2, and the first three terms when c =s 3, and so on. —
Except however the cases in which m is some negative num-
ber equal to or less than c ; in which cases the divisors, m -f<,
m + e-l,m-fc - 2, Sec, becoming equal to nothing, be-
fore the multipliers c—1, c— 2, &c, the corresponding terms
of the series, being divided by 0, will be infinite : and then
the fluent is said to fail, as in such case nothing can be deter*
mined from it.

61. Besides this form of the fluent, there are other me-
thods of proceeding, by which other forms of fluents are
derived, of the given fluxion z = (« + x n ) m oF a ^ l i, which
are of use when the foregoing form fails, or runs into an in-
finite series ; some results of which are fjfiven both by Mr.
Simpson and Mr. Landen. The two following processes are
after the manner of the former author.

62. The given fluxion being (a + jr«)"»j^*- 1 < i ; its fluent
may be assumed equal to (a + x")"^ multiplied by a general
series, in terms of the powers of x combined with assumed
unknown co-efficients, which series may be either ascending
or descending, that is, having the indices either increasing or
decreasing ;

viz, (a+x»W x (Ax r + Bxr~* + ex*-* + dx*-* + fcc),
or (a-for*)"* 1 X (ax* + ax*** + cx*>* + vx r + 3 * + kc).


Digitized by



And first, for the former of these, take its fluxion in the
usual way, which put equal to the given fluxion (a + x»)»>
a<*- l i, then divide the whole equation by the factors that
may be common to all the terms ; after which, by comparing
the like indices and the coefficients of the like terms, the
values of the assumed indices and coefficients will be deter-
mined, and consequently the whole fluent. Thus, the former
assumed series in fluxions is,

n(m+ l)*"-*i(a + *»)■» X (a** + B**~ + c*r-* fee.) +.
(e +x n y n * 1 i X (rAJC** 1 + (r— «) n**-*- 1 + (r -2«) c* r ~*»-i
fcc) ; this being put equal to the given fluxion C a +* B ) m *° , - 1 .r,
and the whole equation divided by (a+x*}*x~*£, there results
ji(«+1)x» X (a*** + B*« + ex?*** + oar r *-* 3 '+&e) >
+(«+**) x(rAar r +(r-#)Ba?^+(r-.2t)cj?»^fcc)5 — **"'
Hence, by actually multiplying, and cedes it in the series, by substituting for y as before.

Thus the fluent ot-^-=± *«- *,(,. + *).-■=*

76. In like manner we may proceed for the series of simi-
lar expressions where the index of the power of x in the nu,
xnerator is some odd number.

Ex 1 To find the fluent of — f**. Vx . Here assuming

va* 2 -*-* 8 ) .
y as ^ */(** + a * )» and t"*"* tne fl uxlon » onc P 8 "' 1 of *J
will be similar to the fluxion proposed. Thus,y = %x±

a/(x* 4- a 2 ) 4 f** T , ; hence at once the given fluxion

^^ s= y — 2o:i V( xS + a * )» theref.the required fluent

by the 2d form of fluents.

Digitized by

Ex. e.



£x. 2. In like manner the fluent of — ~ — -,is

X 2 ^(x^ — a 2 ) — *(.r 2 — a* )*.


^x. 3. And the fluent of ( g * = — x 2 ,/(a 2 — x* )
^x. 4. To find the flu. of ,*[* . from that of ***

Here it is manifest we roust assume y c= x 4 \/{x 2 -fa* j #

. _« _. OXS& 4u*£*3i

This in fluxions and reduced gives , = ___+_£-_
and hence ^g— - «fc - «*- ^ ( g__ ; and the flu.

the fluent of the latter part being as in ex. I, above.
In like manner the student may find the fluents of

,. ? • — •> anc * —nr—r^ He may then proceed in a similar

way for the fluents of — 2L5_ _ * * QN , &c, -— ^^^ r - r
7 Six* pfca 2 ) >/(j? 2 ***) */(.r*;fca 2 ) '

where » is any odd number, viz, always by means of the
fluent of each preceding term in the series.

- 77. In a similar manner may the process be for the fluents
of the series of fluxions,

x x'x x* x & ■***

. y/{a ;&*)' y/^a&c)' y/(a&x? ' ' ' * </(a&x?

using the fluent of each preceding term in the series, as a
part of the next term, and knowing that the fluent of the

first term — ^-7- — w given, by the 2d form^f fluents, =

, •at*

2\/(a$;x) y of the same sign as x.

. Ex. 1. To find the fluent of — — -— x , having given that

of— ^ — - = 2\/(x+a) = a suppose. Here it is evident

we must assume y =x^/(x+a)^ for then its flux, y a= ■■■

hence ^ ■ = *« — JaA ; and the required fluent is Jy—

• Jaa = \x s/ (x+a) - |a V (*+«) •■= (*-««) X |%/ (*+«).
In like manner the student will find the fluents of

** -and **


Ex. 2. To find the fluent of , , having given that
of —i^ — \-z b. Here y must be assumed => x* Wx+a} ;
for then taking the flu. and reducing, there is found * - =

\y - fii •, theref. / -^g^ » fy - **B » fx* V(r + a)

— | <m « jx* \/(* + <*) — #* C* w 2a) x | V (* + a ) =»
(&x* - tax + 8a») X t| V (* + «)•
In the same maimer the student will find the fluents of

f* x z add of -£ — y And in general, the fluent of *
being given = c, he will find the fluent of —. x = ^-xt

x» t/(* + a)- 2^T flC -

78, In a similar way we might proceed to find the
fluents of other classes of fluxions by means of other
fluents in the table of forms; as, for instance, such as
x x *S(dx— a^)>« 8 i\/(rfj:— ^), x3iy^(dx— j: 3 ), fcc, depend-
ing o» the fluent of x% /(dx— **)> the fluent of which, by
the 16th tabular form, is the circular semisegmcnt to diame-
ter d and versed sine x, or the half or trihneal segment con-
tained by an arc with its right sine and versed sine, die dia-
meter being d.

Ex. 1. Putting then said scmiseg. or fluent of xj(dx -x*)
-am a, to find the fluent of xi^{dx-~x % . Here assuming

y » (dx ~ x*)*, and taking the fluxions, they are y =
\(dx - 2xi) U (dx—x*) ; hence x x V ( rfj? -■*•)-» J^iV
(dx - or 8 ) — |y is* |di «— $£ j therefore the required fluent,

fxxy(dx— **), is j<*A-*y =* JtfA—'Kdx — **)*■* b suppose,
-fix. 2, To find the fluent of x*£J(dx— x*), having that
of xivfcLr— xf) given *s B. Here assuming y— x(dr-ix»),
then taking the fluxions, and reducing, there results y =
({^j-^y^ - •**) ; hence x*i \f {dx — x a ) as \dxi
V<dx-x*)~&« f^i— Jy, the flu. theref. of x*x<S(dx — x»)

ia^B — Jy = |r/B -£x(dx— x*)».

-£*. 3. in the same manner the series may be continued

to any extent ; so that in>general, the flu, of x"~V (dx — ' x a )
being given » c, then thfe ne*t, or the flu. of x n x V (<** — x 9 )

79, To find the fluent of such expressions as *

. <s(x*dz 2ax)>

a case not included ra the table of forms. Put

Digitized by





Put the proposed radical y/(x*± 2*r) = 2,orr»± 2ax
z=x z* ; then, completing the square, ** ± 2a* -f c a =- z 2 +a%
and the root is x ± a = s/ (z* + a*). The fluxion of this is

* ~ 7W+lFy thercf - ^(^V^ = 7W+^) ; the

fluent of which, by the 12th form, is the hyp. log. of z+ */
(2* +a*) ^hyp. log. of * ± a + \/ (* a ± 2«x), the fluent

-E*. 2. To find now the fluent of ** ■ > , having

given, by the above example, the fluent of *.

suppose. Assume y< (*a +2o*) = y ; then its fluxion

Xx T oi . # . - jpi . i ' .

V^f +2ox ""* * F * v(* s + 2ax) = y ~ •(** +2ojf) ^ *
— «a ; the fluent of which is y — a* = v (** + 2<w) — a\
the fluent sought. f

Ex. 3. Thus also, this fluent of ^ being given,

the flu. of the next in the series, or *** - — «ui be found,

v'Or* +2<zx) '

by assuming x</(x % +2ax) = y ; and so on for any other

of the same form. As, if the fluent of ■ ,**~ lj L,- be given

= c ; then, by assuming *»~V(** + *«) -» y> the fluent

of 5^5=0 - ^ v( ** +««)-" *«•

-Ex. 4. In like manner, the fluent of -77— -* bebg

j?ireD, as io the first example, that of ; , ** - - may be

r j . x ' — 2oaf) '

found ; and thus the series may be continued exactly as in
the 3d ex. only taking — 2a* for + 2ax.

80. Agam, having given the fluent of -— i -, which.

x s /{2ax-x^)

is — X circular arc to radius a and versed sine x, the fluents

rf ^£c^'7(£^i)' &c - -^(£:^ n » a y b ° ■"tew*

by the same method of continuation. Thus,

.E** i. For the fluent of —r—^r, assume v'(2ax-^s)

=s^y ; the required fluent will be found - - y/(%ax—3? ) + a
or arc to radius a and vers. x.

Ex. 2. In like manner the fluent of - * * „i is


Digitized by



where a denotes the arc mentioned in the last example.

Ex. 3. And in general the fluent of — -7- — * - is

^/(2<ix — x»)

— TLac— — x"- 1 %/(%ax —x s ), where c is the fluent of
n n

— fl^lf — __ the next preceding term in the series.

81. Thus also, the fluent of x y/ (x— a) being given, =

|( x — a)*, by the 2d form, the fluents of a:i/(x- a),
x*r\/(x— a), &c . . . x n x \/ (x — a), may be found. And in
general, if the fluent of x nr ~ l x'/{pc — a) = c be given \ then.

by assuming x n (x—a)* = y, the fluent of x"x v/( x — fl ) * s
2 , * Sim

82. Also, given the fluent of (x — a) m x which is — -p-

(*— ay** 1 by the 2d form, the fluents of die series (x— a^x^,
(x— a) m x*x. &c • - -'(x— «) m x«x can be found. And in ge-
neral, the fluent of (x— a)****" 1 * being given = c ; then
by assuming (x —«)"** l x n = y, the fluent of (x — a)"x"x is
* . x»(x— a^M-nac
found *= -tttt •

♦Also, by the same way of continuation, the fluents of
x B x •(« A *) and of x^x (a A j)" 1 mav De found.

83. When the fluxional expression contains a trinomial
quantity, as \/(Jb + ex + x*), this may be reduced to a bi-
nomial, by substituting another letter for the unknown one
x, connected with half the coefficient of the middle term
with its sign. Thus, put z = x+$c : then z* s= x* + cx+ it* ;
tneref. z* - ic* = x* + ex, and z* + 3 -ic» = x* + ex + 6
the given trinomial, which is = z 2 + a 2 ? by putting a 2 =■

JBjc. 1. To find the fluent of — . ■■


Here z = x + 2 ; then z* =» ** + 4x + 4, and z a + 1 =
5 4. 4x + x* , also x = * ; theref. the proposed fluxion re-

duces to '-—.-—; the fluent of which, by the 12th form in

•(l+s 2 )
this vol. is 3 hyp. log. of z + */ (1 + z) = 3 hyp. log. x -f- 2

+ \/( 5 + 4 *+* a )-


Digitized by




£ x . 2. To find the fluent of i V <* + Cx + *»*)=i v' <* X

F • £+7* + *»>

Here assuming a? + ^ = z j then i = i, and the proposed
flux, reduces to iy^XVC* 8 + 7~4di) =s= * V*< <• (z* + a 8 ),

h r2

putting a 2 for — — -— ; and the JJuent will be found by a sim-
ilar process to that employed in ex. 1 art. 75.

Ex. 3. In like manner, for the flu. of x*~ x xs/{b + cxn 4!
c . . I

<Jac ln ), assuming * n + — =r 2, *ia?*~ l x =* i, andx"-^ a —i ;
hence a 8 * + -x» + — - = zK and v' (<***" + cx n + 6) =

X V (z 8 ± «*), putting ± a 2 = -— — ; hence the given

fluxion becomes - i %/ d X \/ (z* ±a*)» an ^ * ta ^ ucnt as in
the last example.

Ex. 4. Also, for the fluent of , ^^ -s ; assume

1 b+cx+dx*

* n "*" id *■ Zj tlien the ^ uxion ma y be reducc< * to the fQrm

-r X —^ — , and the fluent found as before.

an x* Hha*

So far- on this subject may suffice on the present occasion.
But the student who may wish to see more on this branch,
may profitably consult Mr. Dealtry's very methodical and
ingenious treatise on Fluxions, lately published, from which
several of the foregoing cases and examples have been taken
or imitated.


84. Maximum, denotes. the greatest state or quantity
attainable in any given case, , or the greatest value of a vari-
able quantity : by which it stands opposed to Minimum, which
is the least possible quantity in any case.


Digitized by



Thus, the expression or sum a* -f*.fa?, evidently increases
as x, or the term bx, increases ; therefore the given expres-
sion will be the greatest, or a maximum, when x^ the
greatest, or infinite : and the same expression will be* mi-
nimum, or the least, when x is the least, or nothing.

Again, in the algebraic expression «* — bx y where a and b
denote constant or invariable quantities, and x a flowing or
variable one. Now, it is evident that the value of this re-
mainder or difference, a 2 — bx 9 will increase, as the term Ar,
or as x y decreases ; therefore the former will be the greatest,
when the latter is the smallest ; that is a* — bx is a maxi-
mum, when x is the least, or nothing at all ; and the differ-
ence is the least, when x is the greatest.

85. Some variable quantities increase continually ; and so
have no maximum, but what is inGnite. Others again de-
crease continually ; and so have no minimum, but what is of
no magnitude, or nothing. But, on the other hand, some
variable quantities increase only to a certain finite magnitude,
called their Maximum, or greatest state, and after that they
decrease again. While others decrease to a certain finite
magnitude, called their Minimum, or least state, and after-
wards increase again. And lastly, some quantities have sev-
eral maxima and minima.


Thus, for example, the ordinate bc of the parabola, or
such -like curve, flowing along the axis ab from the vertex a,
continually increases, and has no limit or maximum. And
the ordinate 6 r of the curve efh, flowing from e towards
h, continually decreases to nothing when it arrives at the
point h. But in the circle ilm, the ordinate only increases
to a certain magnitude, namely, the radius, when it arrives
at the middle as at xl, which is its maximum ; and after
that it decreases again to nothing, at the point m. And in
the curve noq, the ordinate decreases only to the position
op, where it is least, or a minimum ; and after that it con-
tinually increases towards q. But in the curve ksu Sec, the
ordinates have several maxima, as st, wx, and several mini-
ma, asvu^YZ, Sec.

51. Now

Digitized by



•86; NdwPbtpause the fluxion of a variable quantity, is
the rate of its indrease or decrease : and because the jpax-
imum or minimum of a quantity neither increases nor de-
creases, at those points or states; therefore such maximum
or minimum has no fluxion, or the fluxion is then equal to
nothing. From which we have the following rule.

To find the Maximum or Minimum.

ST. From the nature of the question or problem, find an
algebraical expression for the value, or general state, of the
quantity whose maximum or minimum is required ; then
take the fluxion of that expression, and put it equal to no-
thing ; from which equation, by dividing by, or leaving out,
the fluxional letter and other common quantities, and per-
forming other proper redactions, as in common algebra, the
value of the unknown quantity will be obtained, determining
the point of the maximum or minimum.

So, if it be required to find the maximum state of the
compound expression lOOr — Sx 2 ±<r, or the value of x
when lOOx — 5x* ±c is a maximum. The fluxion of this
expression is lOOi — XOxx = : which being made = 0,
and divided by 10i, the equation is 10 — x = ; and hence
ar= 10. That is, the value of a: is 10, when the expression
lOOr — 5s* ±c is the greatest. As is easily tried : for if 10
be substituted for x in that expression, it becomes ±c+500 :
but if, for :c, there be substituted any other number, whether
greater or less than 10, that expression will always be found
to be less than ± c + 500, which is therefore its greatest
possible value, or its maximum.

88. It is evident,- that if a maximum or. minimum be any
way compounded with, or operated on, by a given constant
quantity, the result will still be a maximum or minimum.
That is, if a maximum or minimum be increased, or de-
creased, or multiplied, or divided, by a given quantity, or
any given power or root of it be taken ; the resuh will still
be a maximum or minimum. Thus, if x be a maximum or

• • x

minimum, then also is x + a, or x — a, or ax, or — , or x*,

or ^.r, still a maximum or minimum. Also, the logarithm
of the same will be a maximum or a minimum. And there*
fore, if any proposed maximum or minimum can bo made
simpler by performing any of these operations, it is better to
-do so, before the expression is put into fluxions.

Vox. II. Z z 89. When

Digitized by



89. When the expression for a maximum or minimum
contains several variable letters or quantities ; take the fluxion
of it as often as there are variable letters ; supposing first one
oftfaemonlytoflow, and the rest to be constant; thenar
other only to flow, and the rest constant ; and so on for all
of them : then putting each of these fluxions =? 0, there will
be as many equations as unknown letters, from which these
may be all determined. For the fluxion of the expression
must be equal to nothing in each of these cases ; otherwise

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