and INFINITE SERIES. Operation it is diminim'd by diminution being too much, it quantity
-7--r
which makes
,
the
it
is
165
fmaller quantity
ftill
which
^;
again encreas'd by the very fmall
too great, in order to be farther di-
by the next Term. And thus it proceeds in infinitum, the Augmentations and Diminutions continually correcting one another, till at lalt ihey become inconfiderable, and till the Series (fo far conminifli'd
01
verted, in
Ids than x, the order of the Terms muft be inihe fquare-root of xx -+- aa muft be extracted as before;
Wh-ii a
12.
Approximation to the Root required.
a lufficiemly near
is
tinued)
which
cafe
is
x
will be
it
-+- 2X
-f-.5
&c.
, '
And
in
Series
this
the converging quantity, or the Root of the Scale, will be -. Thefe two Scries are by no means to be understood as the two different Roots of the quantity aa -+- xx for each of the two Series will exhibit thofe two Roots, by only changing the Signs. But they are accommodated to the two Caf s of to according as a or x may -,
Convergency, happen be the greater quantity. I (halt here refclve the foregoing Quantity after another manner, the better to prepare the way lor what is to follow. Suppofe then fi'-d the value of the Root where we the xx, may y
yv=.cni-\f
11
,wir
or zap -+- pp or
--qq;
= = xx =
Proccfy
;.j
;
yy
aa (If
-+-
XX=
p=
(\f)'
+
-
= -.
'-,
6
oi,'
-
(if
r
=
O-|t.
zaq
(if
+ +
r
rr
rf-f-/)
q}
^ -H^=
2rf?-J-
= aa^-zap ~ xx + = _^
1
VU*
by
-\-pp-,
-{-
?===
or
>
&c. which Procefs
j)
may J
in wo-ds.
be thus explain 'd In order to find
V ua --xx, 1
yy-=aa-\-xx,
=
or
Root y of this Equation wheie a is to be undeiftood as
the
^-f-/', iuppofc y a pretty near Approxii: arion to the value of _y, (the nearer the betand p is the lnv.,11 Supplement to that, or the quantity which ter,) makes it compleat. Then by Subftitution is deiivcd the fir It Sunxx, whole Root/; is to bt fou:,d. plementiil Lqu^i'oa zap -+-//;
=
INOW
as 2uJ>
plement/,) exactly
;-
=
n:iich bigger than ff,
is
v;c : ;
fh;.!l
-f-
-',
za
-
have nearly p
fuppofmg q
(lor
,
to
is
bigger than the Sup-
or at leaft
ve
(hall
have
reprefent the fecoiid Supple-
ment