13b Mr. Hellins on the Rectification 
equal to the difference between the lengths of the infinite arch 
and its asymptote. 
Now, when y = 0, then z =. 0, and u becomes = 1 ; and 
the value of the series, in this case, is the value of d ; and this 
value, if the terms of the series are ranged one under another 
according to the powers of — t will stand as below, viz. d = 
V 1 -j" aa -j- 
A_ 
2 
Yv(iH-gfl) I 
2.4.2 aa [ 
3 5v/Q + a ‘Q 
2.4 6.4^1 
3 S-7\/( 1 + <*«) 
2. 4.6. 8 . 6 aa 
3 A 
2.4.2 aa 
4 - 
4" 
3-5-3 Vf 1 -\- aa ) 
2. 4. 6.4.2a 4 
3-3 7 ■SV'O+tftf) 
2.4 6.8.6.4a 4 
+ 
3-5 -3 A 
2.4 6.4.2a 4 
3-57-5-3v / ( I +-^) 
2.4.6,8.6.4.2a 6 
&c. &c. &c. 
Here, in the diagonal line of quantities which have the com- 
mon factor A, w r e find the geometrical progression 
&c. so that, when a is much greater than 1, this series will 
converge very swiftly. The same progression has place also 
in the perpendicular columns of quantities which remain after 
the diagonal line is taken away; in all which columns we find 
the constant factor </ 1 -f - aa. If, therefore, the sums of the 
very slowly converging numeral series 
3 
2.4.2 
3-5 3 
2. 4. 6. 4. 2 
3-5 7-5-3 
2. 4.6. 8. 6. 4.2 
+ 
+ 
f 
3-5-7 
2.4. 6. 8. 6 
3-5-7-97 
2.4. 6. 8. 10.8.6 
3. 5.7.9. 11.97 
2.4.6.8.10.8.6.4 T 2.4.6.8.10.12.10.8’ 
3-5 
2 4.6.4 
3-5-7-5 
2. 4. 6. 8. 6. 4 
__3v57-9-7*5 
~r 
+ 
+ 
+ 
+ 
3-5-7 9 
, &C. 
2.4.6.8.10.8 
3-5-7-9 1 *9 
2.4 6 8.10.12.10 8 
&C. 
, &c. 
are taken (and they may easily be computed by the method 
explained in the Philos. Trans, for 1798, p. 54 7 to 530,) and 
3-5-7 5-3 A 
2.4.6. 8 6.4.2a 6 
&C. 
