and INFINITE SERIES; 18. Here the Terms of the fir ft or column, remove or deftroy one order, as another, being absolutely equal to noThey bething by the given Equation. ing therefore expunged, the remaining
17,
*3
255
+
_ i_ +a.rj> + _
* ?* w +3 A *"* r
2f,^
v
fl .
ox _ a ll'l
a.\iy
\
)>=o,
-\-axjs-
^
)3
_,;>,
_ "j.
1
|
j
be divided by the comThis being- done, all the Terms Multiplier <?, whatever it is. ftiil be affecled of the third order will by o, of one or more dimenfions, and may therefore be expunged, as infinitely lels than the remain thofe of the fecond order or others. Laftly, there will only
Terms may
all
mon
that
column,
is
i
zaxx
3.vA.'
axy 4- ayx
-+-
Tjy-
=
which
o,
Q^. E. D. in fomething a difbe thus derived, may be any fynchronal Augments of the Let and ferent manner. variable quantities A* and y, as befoie, the relation of which quanwill be the Fluxional
The fame
Equation required.
Conclufions
X
Y
Y
Then may tf-J-X and y 4exhibited by any Equation. be fubfKtuted for x and y in that Equation. Suppofe for inftance 3 o ; then by fubftitution we flwll that x> ax* 4- axy _y tities
is
-
=
-
3 a x .v 4- X a 4-#x.v4-Xx/4-Y have x 4- X y 4- V 1 z 1 ax or in termini* expanfis .v 5 -f- 3X X -f- 3xX -+- X o a 4 aX* ^XY Y 2rfxX -t- axy -\- <?.vY4- aXy -fj 3j 3;'Y 3 o will vaY o. But the Terms ,v ax* -+- axy _y niHi out of the Equation, and leave 3# 1 X 4- 3xX a 4-X 2axX 3
=
|
|
|
3
;
5
3
=
3
=
3
Y- == o, for XY 3/i y* Y of the let their the relation magnitude be contemporary Augments, what it will. Or refolving this Equation into an Analogy, the ratio Y /7X -Lv ,. 2 ..* ?r*-|- ^rX-L. X 1 A of thele Augments may be this, * X =. a* ..v _|-j* r 3.., + Now to find the ultimate rc.tio of thefe Augments, or their ratio when they become Moments, fuppofe X and V to diminil'h till they become vanishing quantities, and then they may be expunged out Or in thofe circumftances it will be of this value of the ratio. 7
aX* 4- axV 4- aXy 4-
,-
,
,.
,
*
-
P
=
y
-
this
-
~^ax
the
is .V
1
fame
,
which
ratio
is
that
as
2f>x--ai
now
.
or
of the .
a 3_)'
the ratio of the
axy
=
l
And
Moments. or
Fluxions,
it
will
be
-
$x-x
zaxx 4- ayx,
as
wss
found before. is no aflumption made, but what is of the ancient and modern both Methods iuflifiable by the received We only defend from a general Proportion, which Geometricians. is undeniable, to a particular cafe which is certainly included in ir. That
In this
way of
arguing there