234 
Proceedings of the Royal Society 
Denoting it by x, we have 
a , b, b, A b, b 
a x + a 2 4- . . . + « 2 + «i 4- 2A + x - A 
^2 ^1 \ 
_ \A, , a 2 . . . . a 2 , cq,‘ A 4- xj 
~ “ K / & 2 & 2 b y’ 
W ^ « 2 > «d A + a;/ 
whence it can be shown that 
^ 2 ) = k( K ^ ^ 3 “ * 
\« 15 « 2 a 2» V W ®1, a 2 * 
and thus we have the theorem— 
&, K b A b. 
\ \ \ 
, «i» A / 
A 4- 
4" tt 2 "b C% + • • • + + «! + 2A + 
* * 
/x( »■ \) 
/ \ A , «l, « 2 «2, «1, A / 
./ xf 5 - j = y 
N/ \a lt . . . . a 2 , aj 
(XIV.) 
17. From (XIV.) it is easy to deduce a series of identities ex- 
in continuants, viz., 
/ 6, h \ l l \ X( Jl ^ ^ A J 2 h l \ 
" \A , «, , a 3 ...ffl 2 , «[ , A/ A) „ 
K / A- *2 V V K f *• :*. *• 6 . M ’ & °- ( V / 
\ a D ^2***®2) %/ v^n ^2 ,,, ®2 » ( *i) 2A , a LJ a 2 ..,a 2J a i ) 
18. With the help of (XIV.) we can also establish an important 
proposition in reference to the well-known subject of the expres- 
sion of the square root of an integer as a continued fraction with 
unit-numerators. The proposition is : — The general expression 
for every integer whose square root when expressed as a continued 
fraction with unit numerators has q 2 , . . . q 2 , q 1 for the symmetric 
portion of its cycle of partial denominators is 
(<Zi j Vi ? • • • 9.2 > fZi) m ~~ ( ~ l) 1 k G?i » 9.2 > • • • 9.2) ( 9.2 • • • 9.2) } 2 
+ K (<?,...&) m - ( - 1)' K (q 2 . . . q x y . (XVI.), 
I being the number of elements in the cycle. 
