of EdinhibTgli, Session 1883 - 84 . 579 
can be greater than A, and that an element equal to A can occur in 
only one position, viz., in the middle of the cycle. Thus — 
J31 = 5 
Lll 1111 J_ 
1 + 1 + 3 + 5 + 3 + 1 + 1 + 10 + .. . 
* * 
This is the phenomenon of “greatest middle.” For convenience 
we may call a cycle in which it appears a culminate cycle. With 
such cycles several very interesting theorems are connected. 
2. The necessary and sufficient condition that the middle element 
q^ o/ the cycle of 'partial quotients shall he equal to the unique partial 
quotie7it is 
= S’.-*-)- 
Whether the cycle be culminate or not, it is known that 
( A , ■ g,-i){(A ■ ■ + (A... q,)} ^ 
(?i • • • + (?i • • •?.)} 
and as (A. . . . q^^i) and .. . q^-i) are mutually prime and their 
co-factors can have no common divisor except 2, it follows that 
, '/ r = an integer = J say. 
tel... + ^ 
But (A, 2 'i,... g,_i) = A fe,... , g.-i) + te2).*-» 9 z-i), 
and 
tel... • T) = T(qi,> . . , ^.-i) + tem... J 
therefore in the case of “greatest middle” we have 
2Afa , . . , + 2fe , . . . , q,_i) j. ^ 
A(2'i,..., 2'.-i) + 2(2'i,.., g,_2) ' 
Atem..*: q.-l) + ^ql,• •‘,qz-2) 
The numerator here, however, being less than the denominator, — for 
even 2(^2 • . • qz-i) is less — must be zero, and therefore 
te2 ) . . . 5 2'«-i) ~ 2(g'j 5 . . . , qz- 2 ) . 
Conversely, if we suppose this to be the case, we can by taking the 
same starting-point show that q^ = A, that is, that the cycle is one 
of greatest middle. 
The condition is thus necessary and sufficient. 
VOL. XII. 
9 
p 
