382 Proceedings of Royal Society of Edinburgh. [ sehs . 
We thus have the theorem : — 
If the irrational expression ( V /A 2 + 4a/3 + A)/2a be representable 
by a pure periodic continued fraction , so also is (JA 2 + 4a/3 + A)2//3 i 
and the cycle of partial denominators for the one is the reverse of 
that for the other. (V) 
With this may be compared the theorem which it is the 
object of M. Crelier’s paper to establish. 
(5) Irrational expressions of M. Crelier’s form, 
( J A 2 + a/3 + A) -r a , 
are not all representable as pure periodic continued fractions. 
In fact, if we have an expression of this kind which is so 
representable, say 
JA 2 + af3 + A = a l_ JL_ _1_ 
a f a 9 + a s + • • • + a n + • • • 
* 
then, t being any integer less than A/a, it follows that 
(«,-*) + ! L L J^+^ + A-at 
* * * d* a n + «i + • • • = a 
* 
J( A - at) 2 + a{fi + 2tA - at 2 ) + (A - at) 
a 
where the period of the continued fraction now begins at a 
different point, hut the equivalent irrational expression is still 
of the form referred to. An example of this is — 
N /5 2 + 3 x 18 + 5 111111 
3 “ + l + l + l + 2 + 3 + 5 + --- 
* * 
Even here, however, the twin expression ( J5 2 + 3 x 18 + 5) -f 18 
is representable as a periodic continued fraction whose period 
is the reverse of that just written, viz., we have 
n / 5 2 + 3 x 18 + 5 1111111 
18 ” I + 3 + 2 + 1 + 1 + 1 + 5 + .-. 
* * 
(6) The one thing necessary to have recalled in connection 
