66 REV. T. P. KIRKMAN ON THE 
25. Our results are now to be brought into one view, 
each with a reference to the article in which it is proved. 
Observe that m>o and that a factorial containing us 
not integer is always zero. 
(4). P(2ve4+1,v) =P(w+1,2) + P(w + 1,3) + P(w+1,4) + 
=P(x+1). 
2e—4 1 
(8). II(2v7-+m+1x)=1(2e+1,2)-2”- eae 
» (5 +1) 
2a—2|1 
(9). PE(4e +m 41,22) =12(4e@ + 1,22) -2 
j2217 
(10). IP?(47+m+1,22)=P(4e + 1,22) 
2 l 2a—2 | I 
x Fe 3 (m+ a teers ad ; (y+ ) : 
#4 [1 je }1 
In (9) and (10) m is always even. 
m 2a—2| 1 
» (¢ +1) 
(11). PR(6e24+m+1,32)=P(67+1,382)-2°- [=3n1 ; 
(m=8n). 
(12). nay 3x) =41°(67+1,32) 
m 2x—2 | 1 
m, ss jaar we Gry) . 
x oe eee oF os cle 
«2—3 | 
nm (+1) 
(13). Bier Le R(2Qe+1,2)- ) 20 Ce oe 
kax=1. kax=0 
x2}1 
(5 +1) 
RR2e+m+1 Aim 2G og yt) 2 ae 
(22) eee 
m+ 
RR(Qa+m-+1,2) =RQe+1,2).2? 2 
jax=1. kaa=0. ES 
(14). IR(Qe +m 41,0) = FRQx+1,2)-gm. (mtd) 
{ea—pir 
> 
m w—3|1 
n (gt) 
— ?R(2x+ 1,2) x2?- yea 
kaxv=1. 
