71 
On approximating to these roots by the known process, the 
quotients 
1 , 5 ; 3 , 1 , 4 ; 3 , 1 , 4 : &c. 
and 5 ; 4 ; 1 , 3 ; 4 , 1 , 3 ; &c., 
are found, whence the two progressions — 
1 
5 
; 3 
1 
4 ; 
3 
1 
1 
6 
19 
25 
119 
382 
&c. 
321 
0 
1 
5 
16 
21 
100 
5 
; 4 
1 
3 ; 
4 
1 
1 
5 
21 
26 
99 
422 
521 
• 
&c. 
0 
1 
4 
5 
19 
81 
100 
If, instead of proceeding 
forwards in 
either of these progressions. 
we compute backwards, 
using only the 
recurring quotients, we pro 
duce the other progression, 
thus 
3 
1 
4 
3 
1 
4 
3 
1 4 
-99 26 
-21 
5 
-1 
2 
1 
6 
19 25 119 
&c. — 
-19 5 
-4 
1 
0 
1 
1 
5 
16 21 100 
The case of the roots being on different sides of the zero, was ex- 
emplified by means of the equation 
7x 2 — 8xy — 1 02tr— 0 
which are contained in the two beaded progression— 
7 2 3 
171 -23 10 -3 1 
-52 7 -3 1 0 
2 3 7 
4 9 31 
226 
&c., 
51 
and that case in which the roots lie equally on either side of the 
zero by the equation 
lQx 2 — 53y 2 —0 
the roots of which are contained in 
99 
&c. 
-43 
4 3 
• 23 7 
10 -3 
4 3 
1 2 
0 1 
4 
23 99 
10 43 
&c. 
It was remarked that the progressions indicate certain peculiari- 
ties in these continued fractions. First, that the order of recur- 
