138 Mr. Hellins on the Rectification 
H. L. of 2, and l for H. L. of a, for the sake of brevity, and of 
facility in the comparison of the result of the present operation 
with one which is next to be made. The multiplication of this 
series equivalent to A, by its proper factor, may stand thus : 
7 ( x “J" 0 “1” 2.4.4a 5 ““ '2.4.6.60” & c * 
x __ 3 . 3-5- 3 3-5-7-5-3 &c 
2 2.4-2aa 2.4.6.4.2a 4 2.4.6.8.6.4.2a 6 ’ 
£(* + 0 — + 
4.4.4a 5 
3-5 
4. + - 3 
* 2.4.2a 5 1 
2.4.4.2a 5 
3-5-3(*-W) 
2.4.6.4.2a 5 
4.4.6.6a 7 ’ 
1-3 
2.4.4.4.4a 7 ’ 
3-5-3 
2.4.4.6.4.2a 7 ’ 
&C. 
&C. 
&c. 
, 3 _5-7-5-3(>-+/) & 
“ 2.4.6.8.6.4.2a 7 ’ txu 
The value of 1 aa also, in descending series, is a -f* 
Ta ~ 7 ^ 7 * "1" ZjjsT 5 ’ & c * an( * multiplication of this series 
by its proper factor may be made as below : 
a -J — 
1 2 a 
1 
2.4a 3 
a , b 
3 — — —f— 
aa * 
+ 
a 4 
2.4.6a 5 ’ 
— - &C 
„6 } 
&c. 
a -J- — — 
» in 
_ ± __ jl 1 
a 2 / 7 * 1 
I 
2.4a 3 
a 
2a* 
+ 
2.4.6a 5 ’ 
&C. 
2.4a 5 ’ 
&c. 
+ - 
* a 3 
+ 3. &C. 
r, &c. 
a 5 ’ 
3 * 
Now if the series - £ + ^ &c - (which is 
