588 
Proceedings of the Roycd Society 
13. The well-known property regarding the quadratic form to 
which the integers belong whose square roots have culminate cycles, 
suggests that the coefficient of m in the general expression for these 
numbers (§ 4) must have the same form. We are thus led to the 
curious theorem: — 
-^/ (d 2 5 • • • j dz-i) = 2(qj , . . . , q,_ 2 ) then (qi , . . . , q„_i) is of the 
form + 2B^ or according as z is odd or even. 
This further suggests the inquiry as to how A and B are to be 
obtained in any particular case, — how, for example, knowing that 
a, 25, c, 2c, 5, 2a, { } , 
is a culminate cycle, we can partition (a , 25 , c , 2c , 5 , 2a) into a 
square and the double of a square. The result is 
(a, 25, c, 2c, 5, 2a) = 2(a, 25, c)2 + (5, 2a)2, 
where the element 2c separates the given continuant into two con- 
tinuants which are B and A. Similarly we have 
(a, 25 + 1,2, 1,5, 2a) = 2(a, 25+ 1)2 + (1, 5, 2a)2, 
(a, 1, 25 + 1,5,1, 2a + l) = 2(a, 1)2 + (5, 1 , 2a + 1)2, 
(a, 1, 2, 2,2, 2a + l) = 2(a, 1, 2)2 +(2, 2a+l)2,' 
(a, 1, 1, 5, 25 + 1, 2a + l) = 2(a, 1, 1, 52) + (2a + 1)2. 
14. Closely associated with culminate cycles are those in which 
the middle element of the cycle is less by 1 than the unique partial 
quotient ; for example, 
, _ 1 1 1 1 1 1 
'^107-10 + 2+T+9 + 1 + 2+^ + ... 
Indeed, the two cases are almost co-extensive with the case where 
the middle element of the cycle of divisors is 2 : the exact state of 
matters being that if the middle element of the cycle of divisors be 
2 the cycle of partial quotients is culminate or peneculminate, and 
if the latter cycle be culminate or peneculminate the middle element 
of the divisors is 2, except in the solitary instance of ^12 where 
it is 3. 
The complete theory of peneculminate cycles can be very shortly 
given after what has preceded. 
15. The necessary and sufficient condition that the middle element 
