592 
Proceedings of the Royal Society. 
and also (§ 10) the initial conditions 
Ct2 1 } /?2 ~ 1 
tts = 2 , ft = 3 . 
From the two equations by division we have 
ft -2 
- 2 - 4 ”h /^2 - 4 
= 1 + 
«e-4 
Therefore when 2 is even 
a,., = (2, 2, . 
^,., = (1, 2, 2, 2,...)s,-, 
where the suffix afteF the bracket of the continuant is used to indi- 
cate the number of elements in the continuant ; and 
when z is odd 
'^ 5-2 = (2 5 2 , 2 , . . . 
ft_,= (l,2,2,2,..ft,_,, 
Putting z==2r and then z=2r + \ we see that 
CL. 2 f — d2r+l Und ft^ = ftr+l 
and that there is thus no need of stating the two cases separately. 
Our result may therefore be put thus : The number of different kinds 
of cidminate cycles having 2z elements is 
( 2 , 2 , 2 ,....), 
or 
and the number of different kinds of peneculminatc cycles of the same 
extent is 
( 2 , 2 , 2 ,....), + ( 2 , 2 ,...),_. 
toiler e y is the highest integer in - 3) . 
PEIVATE BUSINESS. 
Mr G. A. Woods, Dr Eichard Davy, and Dr John Grieve were 
balloted for, and declared duly elected Fellows of the Society. 
