REV. T. P. EIEKMAN ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 257 
only, when only half the axes are’ clear and diagonal. That is, if we multiply this 
number, under by 
we have the full number Rf(r, k)’\ out of Rf(r, k), that are made by scoring one of an 
even number of axes. 
LXI. Adding the expression in LIX. to this product, we obtain 
Ef(r, ^)=2.2,2. !c) 
+-|{2Rf'^*(r, e)„+R^f'(r, e)„+Rf k—e)^} 
p I 
-1 
comprising all the values of e, h, and n, in our register of (1+ e)-partitioned /-gons 
here specified (A>0, e>0, n>2). 
Thus we have determined the six classes of doubly and singly reversibles of art. LII., 
which are not reducible to a polygonal nucleus, and can register them in order as well 
as the twelve classes of reversibles of art. XX., which are so reducible. 
The only numbers in the right member of the above equation contained in our list are 
R-^<^'( 8 , 0 ) 3 = 1 , R|f( 8 , 2)e=l, Rf( 8 , 2)e=l, Rf( 8 , 4), =2, R|f( 8 , 4),=1, 
(XLIV. and XLVI.). Wherefore 
Ef(8. l)=p;-«'“(8. 
EJ‘(8, 2)=iEr‘(8, 
Ef(8, 3) = Ef(8, 2)..^VKf(8, 2)..i^®+i{E{f(8, 2).. 2^^} 
Ef(8,4)=Ef(8,2)..(2^)’'-.A+i{E“'(8. ® ^ 
= 1 , 
= 1 , 
= i, 
)}=i.H-o=i, 
E:‘(8, 5)=Ef(8, 4)..l+i.E|f(8, 4).(l-0") =2. 
This completes oui’ register of reversible partitions of the octagon. Collecting them 
from XXVIII., XXXVIII., XLIV., XLVI., LIII., LIV., LVIII., we add to those 
above written. 
0 ) 3 = 1 , Rf ( 8 , 2)s=l, R:^( 8 , 4),=1, R|«^( 8 , 2),=I, 
R|f( 8 , 2)a=I, R|f( 8 , 4),=I, 
WJ (8, 2)e=I, Rf (8, 4), =2, Rf (8, 2),=1, 
R^‘^'(8, 5)3=1, R^S, 3)3=1, R^‘^'(8, 1)3=1, 
R''‘-( 8 , 5 ) 3 = 2 , R'^'- ( 8 , 3 ) 3 = 2 , 
R“^( 8 , 1)=1, R“^( 8 , 3)=1. 
2 M 
MDCCCLVII. 
