of Edinburgh , Session 1873-74. 
235 
This is established by taking the general expression for every 
such number, fractional as well as integral, viz., 
and proceeding to determine what form for A is necessary and 
sufficient to make this expression integral. The form found is 
and substituting this for A in (a), we arrive at the expression 
(XVI.) after some reduction. 
19. Further, no integer can be found whose square root when 
expressed as a continued fraction with unit-numerators has 
!2i> 9.2 • • • 9A 2i f° r the symmetric portion of it's cycle of partial 
denominators, unless either K(^ 1 . . . ^ 2 ) or K(g 2 . . . g 2 ) be even. 
This is deducible from the preceding. 
20. Many interesting results may also be arrived at in reference 
to the possibility of expressing in more ways than one by a con- 
tinued fraction the square root of any number. 
All that is requisite in order to find as an equivalent for any 
quadratic surd, J18 say, a periodic continued fraction with a 
period of any given number of elements, say 5, is the solution in 
integers of an indeterminate equation of the form 
21. This leads to the consideration of the various identical 
forms of periodic continued fractions, and on this subject much 
may be learned. As an instance, we may show how a continued 
fraction with unit-numerators, such as is found in the usual way 
as the equivalent of a quadratic surd, may always be reduced to a 
periodic continued fraction with only three elements in its period. 
The identity is 
K ( A > 2.;2a---2a;2l. A ) 
K (?,, 2 a • • • 2a; 2i) 
2 K ( 2 ; • • • 2 j)“-(- 1) ! 2 K(?l • • • 2a) K (2a • • • 2a); 
A + 1 , 1 , 
ci- I- 6 -f- b ci-\- 2 A -f- . . . 
* * 
.cl) (-iy-1 K (bc,..cb) 
. c 6) + K (a & . . . c 6) + 2A + . . . 
