of Edinlurgli, Session 1883-84. 
has corresponding to it a penecidminate cycle of the form 
591 
and there are no other penecidmincde cycles. 
The equation of condition P 2-1 + Q^-i = 2P2_2 is inapplicable 
when 2: = 2 or 1 : in the former case we have 
20. The same considerations which led to the theorem of § 13 
give us now the theorem : — 
If (qi , . . . , q^-i) + (q2 , • • • , 9. i) = 2(qi, . . . , q.-s) 
(q^ , . . . , qz_i) is of the form + 2B^ or - IW, according as 
z is odd or even. 
Also, the expressions for A and B present themselves in the 
same simple manner as before : thus for the cycles above obtained 
we have 
21. It only remains now to see if we can ascertain how many 
different kinds of culminate and peneculminate cycles there ought 
to be with a given number of elements. If the number of elements 
be 2^, then in the case of culminate cycles what we have to find is 
the number of solutions of an equation of type A (§ 10) and of the 
(2 — 2)th degree. Let the said number be denoted by and let 
p be used in a similar way in connection with equations of type B. 
Then (§10) we have the pair of difference-equations 
+ 2A - 1 = A + 1 
1 +A- 1 + 1 +2A'*' ■ ■ ■ 
in the latter, the solitary example 
