580 
Proceedings of the Royal Society 
3. If ^A^4-D he expressed as a continued fraction with unit- 
numerators^ and the middle element of the cycle of partial quotients 
he equal to A, then 
tel,. 
We have already seen that generally 
(•'^ , * . . , Iz-i) i Iz) } . 
tei>---» 5^.-i){tei,«..» ^.-2) + tel,..., iz)V 
and J in the preceding paragraph being shown to be 2, we have in 
the case of “greatest middle,” 
(A , 2^1 , . . . , q,-i) = tel , . . . , g ) + tel , • • • , R-^) • 
Hence 
= (2'2,...,2'^-i,A) + A%i,...,g^-i) + fe,---,!Z»-iA) 
H= A2 + 
2(g2,-.-,g^-i,A) ^ 
-1) 
as was to be proved. 
4. The condition (q£ , . . . , qz-i) = 2(q^ , . . . , q^-o) being satisfied^ 
the general expression for all integers lohose square roots have cid- 
minate cycles is 
[((Zi, ... ,3'2 -i)m - (3'i, ... , 2'2~2)(9'2, ••• ,9'2-2)]^ + ( ~ 1)*{4(3 'u jS'a - 2 )^ “ 2(g'2, ... 
From § 3 we have 
T-\ _ ^te2 , * * * , QLz-i , A) 
(?i, •• • . ?.-i) 
2-^((?2 ) * * • . ^ a - l ) ^(^'2 » • • * » _ 2 AQ 2 _i + 2 Qj _3 ^ 

Alultiplying both sides by P ^_2 and using the identity 
V-iQ.-2-P.-2Q.-1 = (-1)'-' 
we have 
„ ^ 2A{P,..Q„3-(-ir'} + 2P.-2Q.-2 
^z-l 
^ 2-1 
