45^ Mr. Hellins on the Rectification 
A = 
B = 
C = 
D = 
&c. 
circ. arch, rad. being 1, and sec. 
\Z(2iu—i) , A 
2«M ' 2 * 
V0«< — l) 
4« 4 
i/(uu— i) 
6« 6 
4. JL 
+ 4 ’ 
+ i£. 
i- 6 5 
&c. 
And, lastly, by multiplying these quantities by their proper co» 
efficients, and collecting the several terms in due order, we 
shall have (Theorem V,) 
% = y/ ( uu — 1 ) 
J2LA4--^-B_ 
2 * 2.4 
3 * 5 “ 
C -f- 
3-5 -7 a 
D, &c* 
z.4.6 1 2.4.6. 8 
Here it is remarkable, that the terms ^ ^6V '~ » 
4<r 
&c. which enter into the values of B, C, D, &c. always decrease 
while y increases from o ad infinitum; and indeed decrease 
more swiftly than the terms of either of the descending series 
in the preceding articles ; and therefore this series may be used 
for computing the length of any portion of the hyperbola. For 
although the terms of it, taken at a great distance from the first, 
will diverge by the powers of aa, when a is greater than 1, yet, 
as the signs of these terms are alternately -j- and , it admits 
of an easy transformation into another series, which will always 
converge by the powers of . It also wants no correction ; 
in consequence of which it has a peculiar use, which will appear 
in the next section. 
8. But the fluxionary expression obtained 
, • « U ® U 
in the preceding Art. is = — — — ~~ Vi uu+aa) 
V(uu + aa) X VC 1 — 
uu 
x : 
1 -f — + 
8 2 uu 1 
2.4U 4 
+ 
3-5 
2.46M 6 
+ 
3 - 5 - 
2.4,6.84 s 
&C. 
Here the fluent 
