234 REV. T. P. KIRKMAN ON THE PARTITIONS 
R%(r, +1)" 
ee 
 Ok-40r—2) +8 
v2 r—x—2) | 1 ; 4(7—w—k—5) | 1 2 
1] i} 
Eee re eet D): 
where & is odd, r and x are even, x0, x}(7-2h-6) 
at(r-k-5). 
This is the number of (4+2)-partitioned r-gons which 
have an undrawn diagonal axis and two marginal faces. 
P(r, &+1)(k odd) 
es aul | sind CPC) ee ee gk-Hr—2) +2 
24x pae-7—) Jl ir —-—-k-5) |1 
k+l Mi k+l [2 
_ Gr 2k-2)) 21 G(r- 24) *! | (art. 18), 
k+l 1 eth 
1? | 13 
where 7 and w are even, x<0, e<t(r—2h-4), e}(r—K&-5). 
This is the number of (4+ 2)-partitioned 7-gons doubly 
irreversible, when / is odd. Their number when £ is even, 
the marginal faces being in either case two, is 
fh 
_ Pr, e+) (i even) 
(3(r-4))#-" Il (2D et -ors m 
ee E peers Th » Qi-Hr—2)+8 
_— 
— Ge | (rt. 14), 
pr 
where 7, & and x are even, v<0, e<+t(r-2h-6), epr—-k-6. 
In all the above formule irreducible fractions are to be 
counted zeros. 
16. The expressions for the (4+8)-partitions of the 
r-gon which have three marginal faces, can be easily ob- 
tained in terms of 7 and & by a process little differing 
from that above pursued. But I do not see how the re- 
