576 / 1042
608
F
L U
x
filY. r.1uhiplY lhe Ouxioo of thc root by the
exrr.n~ot
uf
Ihe pOlOer. aoJ lhe
proJu~
by a power of the 1.1111
ruot
lefs I,y uoityth,lOIhe gÍl'eo
e~pooenl .
The
Oll~ioo
of
x'
is
~X' ;.
of
1"
nx' -' ;;
for tbe
root of
1"
is
x.
~'hofe n~~ioo
is
~ :
whieh Illllltipliecl by
the expooellt ", aod by ¡ powcr of
x
Icfs by
unlly
th.lon,
gives the above fiuxi..n.
.
,
lf
x
rceei"e the iocremeot
x.
it hceoOles
x+t;
r,ife
Mth tOthe powcr of
11,
aod
x'
beconm
,Y'+""
11-'
~+
~\,'-'x'+,
6,.·
but all Ihe parts of the iom·
l
'
meor, exeepl the firn lerm,
~re
owing 10 Ihe accelermd
iocreafe of
x' ,
and fOfOl me,fllre! of Ihe higher fluxioos ,
The Grn term onl)'
me~lum
the (¡rn ftuxiun; th. Auxton
,
'
___ T
.
T
of
~ '+z '
.is
~x~%Zx.' +z '
;
for pnt ..,=.I'+Z', we have
•
•
t
x=~:: ,
and Ihe fiuxioo of
x
T ,
which is equal 10 Ihe
propofeJ Buen!, is .} .:-
x,
for whieh fubniluting the n lues
or
x
aod
x,
we hHe the aboYe fluxioo.
RULE
11.
To find the fl,txion of the produ[t of fc·
fecal "ariable quantit,es multiplicd togcther, multiply the
Auxton of mh by the produd . f the ren of the qUAn·
tiliel, anJ Ihe fum of Ihe produal lhus ¡"fiog will be Ihe
./luxion foughl.
Thus the fiuxion of
XJ,
is
;J+)~:
Ihat of
x)'t.
is
X)~+~%J'+)Z;;
¡nd Ihal of
X):/I,
is
xJ%d+xJllz'+xzllj
+J7.ux.RULE
III.
Tu 6nd fh efiuxion
oC
a Craaioo.-From
the Auxion
oC
Ihe numCl210r multiplico by Ihe deoomina·
lor, fublraa lhe ftuxion
oC
the denonlinatQr multiplied by
the numerator, anddivide the remainder by Ihe fquare af
me deDoOlioator.
Thus, the Buxionof!... is )
Y~xJ
¡ tlut
oC _+x_
,
ís
___
j)'
x
J
;x
x+J-~+jxx
)..
~-.Y"
-;tj)-'
-=;;:j'"
R UL~
IV. In
e~mrlex
c;¡fes, let fhe
p~rticulm
be
colleaed from the fimple rull!
,no
comb,ocd togtther.
.
x';'
~X,),; +,.'txX%-\ 'l' Z
The
flu~ton
of -ís -
~
'/_-- _. .
Cor
z
z'
,
lhe fluxioo of
x'
is
,.~, a~d
of
i
is
7J;,
~y
Rule
l.
and lhcrcfore Ihe Buxion of
1') '
¡by Rule
II )
7l '))+
:/' x;;
from which . nluhiplicJ byz, (by Rul'
11
f. ) .nd
fubl tAaing f,om
11
the 9uxion of Ihe ,knOmlO2l0r
%.
mtl
riplled by the numcr.lOr, and divldlog the
\\h~lc
by tbe
fqu~re
uf the dcnumln¡tor, g'vcs lhe above
6l1x.on,RUL[ IV. Tite feeond Auxion is Ikrlvc,l from fhe
6, ,
in the
f.meDl looer as the 6, n
frU")
lit n"wtng
quanlll)'.
.
'J1tUSthe floxionof.y •
31' , ;
i
5
fceoO/I.
6" ' +3~ '~
(b Rule
JI ) ;
.n fu
00:
hut If
1
Le
iO\JllJble,
l=:.J,
':1-<1
the flcond 8uxion of x'=(,y, ' .
r"oe.
l. 7 0
¿""fln;",
""x¡'n'lff!IJ
",i.,imJ
;/!h aqUlnlÍf)' mcrCJf,s, tts
OU~IUD
" puGlIYt ¡ when
o
N
s.
il
rlrcre~fcs.
it
i~ n~gu;ve ;
t"mCnte when il is
jun
be.
tWI~t inernfio~
. ntl deerraling, ill tluxlun
IS
=0,
RULE . Ftnd tite
nll~IOO.
m,lke il
=o,
wh.nee
In
e.
quallon IVill rclult th.t IVdl ci'e ao ¡ofwer10th, qudllun.
F'g,
4.
EXHI' . Tu dClCOmtne li,e dimenhuns o(
1
e)'lioJllc m<afure
AIIC U,
"I'en jt the 10r, \l/hlth Ih .11
ClIllraio 2 givro quantity (of "quur, gra,o,
6•.)
unJer
Ihe leal! Inle,nal luperficitS paffible.
Let lhe diJmeter AB=x, aod the
~hilUde
AD-=,, ¡
mOl eOVer, Ict
f'
(3 ,14 159,
(¡c,)
denote tltcpcrophe,y
oC
the corele whof, d,ammr is UOifY, . "dlct ( be ,hcglr,
n
content of lhe c)'lindcr. Then it wllI be I :
p':
.:
(r .)
the cireumfmnce of the bafe; which. mufuplied by Ihe
.ltitud,
Jo
givo
pXJ
fur rh, conea'e fuperfiClcl uf Ihe ey.
lioder.
In
likenuoner, the am of the balc.
by
mulri.
plylog the fame
exprel~un
into
T
of the diall1C1Cr.t, w,lI
be fouod
=pt'
¡ whieh dmvn into the altitude
¡,
B"es
4
p.
'.'
for the folid content of the c)'lilldcr;
~
hieh bei""
4
•
made:(, the coneave furfaee
pXJ
will be founo
=4~'
It
4(
/,1"
.od eonfequently the whole furC¡ee
=-+--:
IVh"e.
.t
4
f
I
B •
h' h . 4"
.J'n
be'
o t le UXIOO,
\V
te
11 - ;,- T -
log pUt
=O,
\Ve
t_
Ihall Cet
-8
<>4-px
l
=o
¡ aod thmfore
lr=l
j
¡:
(ur·
ther, bmufe
! x
J
=8c.
Ind
/,x'J=4(.
it follo"" lhat
X=11;
wheoce
J
is al(o koo..o. and from .."ieh
11
'p'
pms. that the diammr of the bafe mult
be
jull Ihe
double o( lhe .ltilUde.
F'g.
7,
To fiod rhe longen and Ihoncn orllinms of
~y
curve, DEF, whofe equuio. or Ihe
rd~uun
II'htch
tit.. ordinms bear tO the abfe,flas
IS
known.
M. ke AC the abfcilfax. 2nd CE Ihe ordinm
=,;
take • value , io terml of,. and fioo
Ítl
fiuxlon whlCh
m, kiog
=0, '
ao equluoo
,,,11
"fuh . 'hole rOOIl
~"e
rhe
v.lue
~f
x
when ) is a maXlDlum o' miOlmum.
'fo determine ..heo it il a
ntaximum.nd. 'hen
I
míni.
mum. take the value oe)', wheo
It
í, a Imle mmc than
the root
oC
rhe equ,tion fo (ound, aod
JI
OUf be p rc ..,eJ
whcther it iocreaf" o, dem.fes.
1(
the cquat,on 1m an even number of equ
,1
rnot •
will
be
ndthcr a maximum nor nltoimunl witeD íll
aOXt~n
is =o.
Pl OD. l .
r.
dro'lll o lor.'l rnl/1
'''J
(/Ifl / .
r 'g.
5.
When Ihe abe.1fl, CS of a
c~rff
01 ,. ,
u·
nif,trn,ly from
A
tO
11.
tite mOlíoo of rhe
CUllt
~,II
be
rerarded IC
JI
be cone.'e. antl .ccrlet4!ed il
c"n"~
1)'
IVmls AB; for
I
nrltght lioe
n:
ís d,f",be,1
lo)
,n unl'
fOlm mOIlOO, lud the du iun uf
tll~ C~"(
"
"'1
r
ojtl
ís Ihe r.t'ltc:u Ih, fluxloo of tite
l'n~¡O'. h'c.~rttl
." ..
Il
tleC. ,the Ihe rlor,cnt If
I!
c/lnlluucd
10 ''''••
~ eq:,~IIy
11,
tI.arp
lint. No.. II
S,
or
Ce
he
ti
d,,'"" "
ul
tlor
~jlr,
L'J
",11
he 1),.
iI,~ion
of Ihe
t.ln~1
n,. ,.,1
dr
01
I~t
""
.llI1m. !tn.1 bcc.utfe tite
trr.n~II ·1 ~',
,J, .r<
cq'"
.n~III,,,,
'/r: , ::
e :sr.
,,1..
r""rc
J( l'll.
l',od al"unlt rru(>llrll\1l1l1 10
ti"
a.J,·., r