390 
Proceedings of the Boy at Society 
hier gegebenen abgeleitet werden konnen.” When it is recalled 
that Stern’s memoir extends to 102 pages of Crelle’s Journal, the 
magnitude of the generalisation will be appreciated. 
3. The statement of the special case referred to (which is all that 
is needed for our present purpose) is as follows : — “ The general 
expression for every integer whose square root, when expressed as a 
continued fraction with unit numerators, has , q 2 , qi ? 
for the symmetric portion of its cycle of partial denominators is 
fe,. . q^'K.{q^, . . . , Jj) ^ 
I being the number of elements in the cycle J 
The functional symbol K( ) is explained by the example 
K(a, b, c, d)^ 
a 1 0 
-1 & 1 
0-1 c 
0 0-1 
0 
0 
1 
d 
4. Taking the case then of this theorem where 1 = 2, we have 
1 
^ 
(m>1) 
the asterisks indicating the beginning and end of the cycle. This 
is M. de Jonquieres’ first theorem. 
It is desirable to state it in two parts, viz., (1) where q is even 
= 2k say, (2) where m is even, = 2n say. These give 
* * 
V(2n)2 + 2n = 2N+ + 
* * 
By giving all possible integral values to k, M, q, n here, we obtain 
every number whose square root has a cycle of two terms, and at the 
same time we obtain the said cycle. The condition, m>1, is 
necessary to prevent degeneration of the cycle; if m = 1, the 
cycle is 
