/ 
454 
Mr. Hellins on the Rectification 
which series will converge the swifter the greater y is in com- 
parison of 1, but will diverge when y is less than 1. It also 
wants a correction, (here denoted by the letter d 9 ) which shall 
be given in its due place. This series then, when y becomes 
great in comparison of 1, will converge very swiftly, and be- 
comes useful in those cases where the ascending series above 
investigated fail. 
But, since the value of % may be expressed in another de- 
scending series, it will be proper to consider that also. 
5- The expression is evidently = V(i + 
eeyy 
, which, by converting \/ ( i -f 
eeyy 
into series, making 
i the leading term, becomes x : l + 
— — t-'It-s-j &c. Here, the fluent of 
2.4c 4 j 4 
eyy 
5 2.4.6 e 6 y 6 ~ 2.4.6.8c 8 JLAVJL '^? uuw ‘ V (i + yy) 5 
the first term of the series, is e y' ( 1 + yy) > and, calling the 
&c. A, B, C, &c. 
y 
y 
y 
fluents of 'y/ ^ 1 -|" yy ) 5 y 3 VO+yy) ’ y 5 V(*+yy) 
respectively, we have 
A=H. L 
V(yy+0 
ri(yy + 0 — 1 
B = 
C== 
D = 
&c. 
2 yy 
ri(yy-R) 
4 y + 
V(yy+0 
6 y 6 
&C. 
A 
2 
aL 
4 
s£_ 
6 
and thence, (Theorem IV,) 
‘-/{yy + A 
+-4 A — + 
% 
ze 
2.4c j 
2.4.6c 5 
c 
5 
2.4.6.8c 7 
D, &c. 
:—d. 
which series will converge the swifter, the greater y is in com- 
