68 REV. fT. P. KIRKMAN ON THE 
(21), RRGe+m-+1,82) =R' (6x +1,32) 
aaa a2] 1 
| : (5 +1) n(g +1) 
X27. ett a aay eo 
(22). PR9(6x +m+1,3x) = R%(6e + 1,32) 
2a—2'| 1 
s pa a 1) 
1-2 }1 
(28). IR%(62-+m+1,3x) = 4R3(62 + 1,32) 
m w—2|1 
0 bia ae 
ona imenet? (e+) 
- yori Gee! 
32-3 [1 2v—2 | 1 
m—2 (541 we (5 +1) - 
ee Tee oe eer 
26. If now we desire to enumerate P(7+1), the 
(@+1)-edra on an 2-gonal base haying triedral summits 
or, in other words, to enumerate the triangular partitions 
of the w-gon; we write first, 
(24). P(#+1)=P(7+1,2) + P(#+1,8)4+ 
(25). P(jw+1,f)=R(27+1,f)+R(e+1f)+R(e+Lf) 
+P(@+lLf)+P(e+1Lf)+l(e+L/f). 
Ri(v+1,f)=R*(2f+4 (w-2f) + Lf) is given by (20); 
Rw + 1,f) =R(+ (w-2f) +1,f), by (15); 
R(w+1,f)=(RR+ RR*+ RR’) (2/4 (@-2f) +1,f), 
by (18, 17, 18, 21). 
T(e+1,f) = (114+ 1? + I+ IR + IR? + IR*)(2f+ 
(w-2f)+1,f), by (8, 10, 12, 14, 19, 23); 
P(w+1,f) = (EP + PR 2+ (v-2f) +1,/), by (9) 
and (16); 
Met CP+TENAt+ (2-9) +12), by OD 
and (22). 
And thus P(#+1) can be found in terms of 
2 /P(f+ Lf)=2 (B+ R?+R4P4+P4+DH(Q+LS), 
where f has every value from 2 
to the integer next below 
da. 
