54 $ JMtr. Hellins’s improved Solution of 
not pursue this method any further ; but, having examined his 
process, and corrected the errors of the press which occur in it, 
now give the equations expressing the values of C, D, E, F, &c. 
which were obtained by that method. 
so. For the sake of brevity, let y = d; then will the general 
values of C, D, E, F, &c. be expressed by these equations : 
2 n 
A — 2 
d 
B 
n — 2 
(n 
+ i)B- 
4-dC 
n — 
3 
{n 
+ 2)C 
- 6 dD 
n — 
4 
( n 
+ 3)D 
-8rfE 
n — 5 
&C. 
where the law of continuation is very obvious. And the parti- 
cular values of these letters, when 71 = f, -J, and f, will be as 
expressed in the following columns : 
±L 
3 
8 d 
5 
12 d 
7~ 
1 6 d 
B — 4“ A 
3 
C — ^-B 
D — — C 
7 
— D 
9 
E 
&c. 
n — 
3 
2 
B - 
i 
6 A 
i 
o 
00 
3 
3 
— D — 
ic 
5 
5 ■ 
— E — 
— D 
7 
7 
&c. 
4>d B -j- i o A 
— c — ^ 
l 
i 
— D- 
_9_ 
3 
3 
1 6 d -p* 
1 1 
C 
D 
21 . The solution of the problem being now finished, it may 
perhaps be satisfactory to the reader to see how the sums of 
the very slowly converging numerical series, which arose in 
Art. 7 , ii, and i 6 , were obtained ; the investigations of which. 
