590 
Proceedings of the Royal Society 
18. In a penecidminate cycle of 2^ elements q^, q2, &c., ice have 
90-1 = 1} oind for the dMermination of the others 
(?1, • • • , = ■ ■ ■ , ?.-s) + (?2. • • • . ?.-2) + (?2> • • • ?. s)- 
From § 15 we have 
2fe , . . . , ^.-2) = fern • • • , ^0-i) + (^2 ’ • • • j 1) 
= QzMi j • • • 5 P-2) + fei , . . . , qz-i) 
+ j • • • ) ^70-2) + fe 5 • • • 5 70-3) 
7z-i<2 
= 1 
and . •. 
(7m • • • j 70-2) = (7m • • • j 70-3) + (72 » • • • 5 70-2) + (72 ^ • > dz-^h 
19. As the equation here got for the determination of 
7i > 72 5 • • • 5 70-2 the type B formerly investigated in con- 
nection with culminate cycles, it is evident that having got all 
possible culminate cycles of 2z elements we can at once write down 
all possible peneculminate cycles of 2^ — 4 elements. Thus sup- 
pose it be required to find all the peneculminate cycles of 12 
elements. Eepresenting the cycle by 
a, 5, c, c, { },.... 
we know that e = 1 and that for the determination of a, h, c, d we 
have 
(a, b, c, d) = {a, h, c) + (5, c, d) -i- (5, c). 
The solutions of this equation, however, have been found in § 12, 
in considering the culminate cycles of 16 elements, to he 
a, h, c, d=\ , 5 , 1 , 25 -f 2 
= 25 + 2,1,5,1 
= 2 ,2,2,2. 
Thus all the peneculminate cycles of 12 elements are 
1 , i, 1 , 25 + 2, 1 , { } . 
26 + 2, 1 ,6,1 ,1,1},... 
2 ,2,2,2 
The connection between the two kinds of cycles may be best 
formulated thus : — Every culminate cycle of the form 
fi,l,7r,p,(T,...,a), 2n + 1 , { j , . . . . 
