45® 
Mr. Hellins on the Rectification 
u uu 
= “ + + 
V(« + — 0 
VO- 
u* 
2 U* 1 2-4 It 8 
3-5« 
2.4.6 1 ' 
+ 
3-5-7« 
2.4.6. 8 m i6> 
&c.; the correct fluents of which are (1 heorem VII,) 
r 
3-5 
3-5-7 
24.7M 7 
2.4.6. 1 Hi 11 
2.4.6.8.I5Z4 15 
■, &C. 
[ — - a. 
Which series is better adapted to this case than either of the 
preceding ones, in that it is much simpler, and converges twice 
as fast. And the correction of it is easily attainable by various 
methods. 
10 . But the original fluxionary equation in Art. 1 , admits of 
a conversion into series, two different ways from any of those 
which have yet been taken. For, by the Binomial theorem, 
aa yy \ ... i aa y yy a ‘ * 7’ V i 3 a 
*=74 
yy 
‘ +yy 
3-S a 
+ 
24.6.8 * (i-f;y;y) 4 
for the fluents of y—y, jr+yy) 
have 
1 +yy 2 -4 ’ O+xy)* 5 2 -4- 6 
j yS ■, &c. where, putting A, B, C, &c. 
- , — ( , &c. respectively, we 
yy 4 
A =y — circ. arch, rad. i , and tang, y. 
B= — 
B = — 
&c. 
2 (i+w) 
y 
4( I +3’4') Z 
V 
6 (H4J) 3 
&C. 
4 
4 
4 
jA 
2 
jL 
4 
7 C 
6 : 
and thence (Theorem VIII,) 
%=y 4 
A 
2.4 
-B + 
$a 
2.4.6 
MfL D, &c. 
2.4.6 8 
In which series, it is pretty evident, the quantities A, B, C, D, 
&c. will have a convergency while y increases from o ad infi- 
nitum, although the convergency will be but slow . after y 
becomes greater than 1 . It is obvious too, that this series 
