397 
of JEdinlurgh, Session 1883-84. 
M must both be even : putting therefore q—^'p and m = 2n, we have 
as our final result for the case 
^|(4^2+l)N+J,p + 4i.N + l={ }+Yp+Y^ + f-}+ .. 
* * 
This furnishes a theorem closely resembling that in § 9. 
here M. de Jonquieres’ ratio — 
2(4p2 ^ 2p 
~ 4pN + 1 
2n 
For 
^2J9 + 
Hence, 
4pN + 1 
1 
2n 
If the periodic continued fraction for JAd' + d^ (d prime to A), 
he wanted, and the continued fraction equivalent to 2 A -^d he found 
to he of the form 2p + 2 ^^ the periodic continued fraction 
. ^ . A 1 1 1 
required A + ^ - — -~ 
^ 2j9 + 2j9 + 2A + . . . 
* * 
13. These two theorems, as might be inferred, are not isolated 
from the main subject ; and it is of importance to see their exact 
position in the theory, not merely on their own account, but because 
this can be done by establishing a general theorem, which affords a 
complete solution of the problem M. de Jonquieres set himself, viz., 
to find what relation exists between the ratio 2a : d and the terms 
of the cycle. 
We have seen that 
2a = K(gi,22, . . . , ?> - ( - ri%i , gj, ■ • . S 2 ) 
and writing c for ( - 1 )'+^K (^2 ? • • • > ^' 2 ) 
MK(gj , 
. .. 
, gi) + cK(g,, . 
A 
1 
0 ... 
0 0 
0 
-1 
$2 
1 . . . 
0 0 
0 
0- 
■ 1 
^3. .. 
0 0 
0 
0 
0 
0 ... 
9.2 1 
0 
0 
0 
0 ... 
-1 9x 
c 
0 
0 
0 ... 
0-1 
M 
