and INFINITE SERIES. and IK
275
=
X, will continually decreafe, and by decreafing will apand K, and nearer to the Ratio of the Velocities at nearer proach and will finally vanifh at the fame time, and in the proportion of
G
GN
to KN. thofc Velocities, that is, in the Ratio of Confequently in the Equation now form'd, if we fuppofe to decreafe and and at laft to vanifh, that we obtain their ultimate continually, may
X
Y
GN
mail thereby obtain the Ratio of to KN. But when and vanifh, or when the point F coincides with G, and I with .x'; fo that we fhall have H, then it will be y, and : x :: And hence we mall KN. obtain a Fluxional Equay tion, which will always exhibit the relation of the Fluxions, or Ve-
Ratio
we
;
X
Y
EG
GM for
Thus,
Lines y and
may
y*
o
= Y
;
and x
(fee before,
aX
= IK =
if EF=j', and HI x, and the indefinite are fuppofed to increafe at fuch a rate, as that their
1
-f-
for
we
pag, 255.)
axY
-\-
aXj
which may be thus exprefs'd 1 aX .+. ay +- 3tfX -h X Analogy, when
Y
X
and
Ratio, will become becaufe it is then
Y
Y
:
X
this
FG=Y,
X
-+-
1 Equation x
ax* + axy X, by fubftituting x, and reducing the Equation that will
always be exprefs'd by then making
for j,
zaxX
=
Example,
A:
relation
arife,
HK
belonging to the given Algebraical or Fluential Equation.
locities,
y -f-
=
:
have ^x"-X
fhall
-+-
rfXY
in
an Analogy,
^
:
and
-f-
3y*Y
Y
Z
X
-f-
Y X :: 3** 37 Y -+- Y*. 3jyY*
:
aX -+-
ax
3#X
!
3
-
=
o,
2 ax
This
are vanishing quantities, or their ultimate And ax. ^ax -f- ay 3** 3^*
X
:
:
:
GN
KN
x :: x, it will be y the of the Which gives ax. -+- ay 3X 3^* proportion And the like in all other cafes. Q^. E. I. Fluxions. might alfo lay a foundation for thefe Speculations in the fol1
zax
:
::
:
::
v
:
:
:
We
lowing manner.
Let
ABCDEF,
6cc. be the of a Polygon, Periphery or any part of it, and let
the Sides
AB, BC,
CD, DE, &c. be magnitude
of any
whatever.
In the fame Plane, and at
any
diftance,
draw
the two parallel Lines /6, and bf\ to which continue the right Lines
BCcy, DEes, &c. meeting the AB4/3,
parallels as in the figure,
N
n 2
Now
if
we
fuppofe