1902—3.] Dr Muir on Pure Periodic Continued Fractions. 381 
that a continuant is unaltered by reversing the order of its diagonal 
elements. We are thus led at once to the results 
1 
a, + . 
* 
+ a n + 
* 
1 
a n + , 
* a n - 1 ' 
+ cq + 
* 
, . . . , q w _i) , (H) 
(a 2 j • • • i ) 
(a l , . . • , a n - 1 ) • ( a 2 » • • • ’ a n-i) 
{a n a >2 ^ ~ 1 ) 
1 _1_ 
4- « . . ! 
Cl. + 
*2 
Cl,, + 
1 
1 
(HD 
a n _i + • • • + ci, 2 
(3) From (I) it is easy to formulate the conditions which must 
hold in order that a given irrational expression ( J H + R)/D may 
be transformable into a pure periodic continued fraction: it is 
more convenient, however, to put this quasi-converse proposition in 
the following form : — 
The irrational expression ( N /A 2 + 4a/3 + A)/2a is transformable 
into a pure periodic continued fraction , if positive integers a^ , 
are determinable so that 
. , a, 
and 
A _ (a T , . . . , a n ) — (a 2 , . . . , a> n -i) 
a (a 2 , . . • j a n ) 
P = (a 1 , . . • , a w _i) 
a (a 2 , . . • , a n ) 
(IV) 
(4) Tf these conditions he satisfied we obtain by division 
A (oq , . . . , ~ > • • • > ^-i ) 
j 8 (ci]_ , ■ . . , 
and therefore 
*v/A 2 + 4ctj8 + A 
2j8 
/ 7 K 
1 ^ | ( a 2» 
...) ) j (®i ? 
vl 
2(a 1} J ( a i> 
2(a 1 ,...,a n -i) 
{VKv 
■ , «w) - ( a 25 • • ■ > + 4 ( a l ’ • ' : 
l)(d2» •••> 
*) + {( a lJ — ( a 2’ ■ 
2(Ol, 
. . . , CCn—l) 
l 
1 
: n n +- 
* 1 
+ • • • +(*!+•• 1 
