1902-3.] Dr Muir on Pure Periodic Continued Fractions. 383 
with the whole matter is the following theorem,* viz. : If a 
quadratic equation have real irrational roots , and these he trans- 
formed into continued fractions of the type a, + — — 
a 2 + a 3 + * • * > 
or its reciprocal , where a x , a 2 , • • • are positive integers , then both 
continued fractions are periodic , and f the one period is the reverse 
of the other. 
Applying this to the quadratic equation 
ax 2 - 2 Ax - /3 = 0 , 
we first note that its roots are 
sj A 2 + a/3 + A ~ sj A 2 + af3 + A 
— a ’ a ’ 
and that the second of these is transformable into 
- P 
jA 2 + af3 + A ’ 
and we thus find that we need not go further. 
(7) There is an important matter connected with the funda- 
mental result of § 2, which, apparently, has not hitherto been 
noticed. This result, if we write p n for (cq , . . . , a n ) , q n for 
(a 2 , . . . , a n ) , etc., is 
1 ^{'Pn Qn— l) fPn—lQn "h Qn—l) 
jj. 1 + cjg + • • • + a n + • • • 2 q n 
* 
Now the number under the root-sign being non-quadrate, its 
square root must be expressible as a recurring continued fraction, 
and the problem is thus suggested of finding this fraction. By a 
* Unfortunately omitted from most modern text-books. See Serret’s 
Cours d'algebre suptrieure, 4 e ed. (1877), i. p. 49. 
t It would be better to insert here the words “ if the periods be different ’’ ; 
for they may be really identical, the mode of writing in the case of certain 
periods being deceptive on this point. Thus, when A = 7 , a = 2 , 0 = 15, we 
, v/7 2 + 2‘15 + 7 * , 1 , 1 1 
1,aTO 2 , 7+ l + lS + i + -- 
y7*+2-15 + 7_ 1+ J_ 1 l + _ 
15 
16 1 7 
where the periods, although seemingly different, are not really so. A still 
better instance is got from A — 7 , a -= 1 , 0 = 30 . 
