1902-3.] Dr Muir on Pure Periodic Continued Fractions. 385 
7 1 + 
{■*{*!} 
I! 
= 10 , 
1111 
2+3+1 +2+3 
10 + 12 ’ 
(8) Taking the two periodic continued fractions in § 2, and 
subtracting from the first the reciprocal of the second, we obtain 
1 1 
ci-, + 
* a 2 -t- • • • + a n H 
* 
Pn C/ n _i ^11 1 
q n a n + a n _ x h b a l H 
* * 
(VII) 
If with a view to obtaining from this an instance of the equality 
of a pure and a mixed periodic continued fraction we impose the 
condition that (p n - q n -\) / <ln be a positive integer, we are led to a 
result which makes the one side an exact copy of the other, viz., 
Pn ~ q n - 1 
j ^2 — 5 ®n — 1 5 ..... 
qn 
(9) Another matter worthy of attention is connected with the 
theorem* that every mixed recurring continued fraction is equal 
to an irrational expression of the form (U + Y n /N)+W. 
Putting 
1 1 
a-, H — 
a 0 + • • • + a m 
1 1 
+ a x + a 2 + ■ 
* 
- = X, 
• • + a n + 
* 
and 
* i+ i+ ■ 
s . 
Ill 
+ 
rH 1 * 
+ , 
we have 
V ( a l > a 2 » • • 
. , a m , x) 
A — / 
( a 2 J • • 
. , a m , X ) 
x(a x , a 2 , . 
• • , a m) + (« i 
, a 2 , , a m _ x ) 
x(a 2 , . . 
■ j a m) + ( a 2 i 
\ 5 
a 3 3 • • • j a m- 1/ 
_ X7T m + 7T m _i 
, say. 
%Pm + Pm— 1 
If now we insert the value of x given in § 2, writing it for the 
moment in the form ( N /H + R) + D, we obtain 
* Serret’s Cours d'algebre suptrieure , i. (1877), p. 46, or Chrystal’s 
Algebra , ii. (1889), p. 430. 
