122 Mr. Hellins on the Rectification 
series in Art. 808 of Mac Laurin’s Fluxions, his E = a -4 
7 * a 
will become = 1 -\-bb, which is = ee in the notation used in 
this paper ; and his series, 
Nf v A /± I gaR 2 S aC ^gaD & 
2 x V £ "r 2 . + e T 4 .6E T 6.8E T 8.10E’ £Xt " 
(A denoting the first term, B the second, C the third, &c.) 
will become N x 
1 
2C 
3-3 
+ 
3 - 3-5 5 
&C, 
2.2.4** 2. 2.4.4. 6* 5 ”1” 2. 2. 4.4.6. 6. Se 7 
which series, since — is = s, and N denotes the quadranlal 
arch of a circle of which the radius is 1, exactly agrees with 
Simpson’s series. 
And this series will be found to agree also with the value 
of sG in the equation marked (<£) in Art. 9, when u becomes 
= 1. For, in that case, eV = ex denotes the asymptote, and 
H the infinite arch of the hyperbola ; and we have, by trans- 
position, ex — H = eG. And, u being = 1, xA., the first term 
of the series denoted by G in Art. 11, becomes = the area 
of a quadrant of a circle of which the radius is 1, that is, = 
— : B becomes = — = ~ : C becomes = ~ = : D be- 
2 4 2.4 6 2.4.6 
comes '= ^ : &c. and these values being written for 
A, B, C, D, &c. and the whole x e, we have tG = N x G -f- 
+ 77^6 + 8’ &c - Perfectly agreeing with the 
series above stated. 
This series, it is obvious, will converge but slowly when e 
is not much greater, and consequently e not much less, than 1 ; 
that is when 6 is small in comparison of 1. But, in such cases, 
other series which have a good rate of convergency may be 
used, as was shown in my former paper on the Rectification 
