34 THE REV. F. H. JACKSON ON CERTAIN 
Similarly A, may be written 
Daf Mpa Os \ 
(m—2)!) (n)! {1}! (nm)! {O}! 
Dn! ( 1 ¥ 1 \ 
(n - 2)! l Pn? Pn-1 Pn-it Pn? Pn j 
= Dn! at fos! 
(m= 2)! Dn y+ Pn Pn (m— 2)! {2}! 
which is 
since (Pn-1+Pn)pPn is {2}! for the m elements p, py... . + Pn- 
In the expression (23), if we reverse the order of the terms we have 
ai Pn! (2)n—« ie (%)n—s 1 Lah (2)n—1 ; 
ee ee cenou ome: clr eee =) orgs 
and since 
(®)n—1 a 
(mn)! = 
in A,, 
Whee = pala Pa) 
UAE =D Dab pany) bcaeten (ay ee + Py —s41) 
also 
{O}! = 1 
{1}! = pp_s:, (Because the term involving {1}! was derived from B,_,,,) 
{2}! = (Drdsso Fh Pig) Peer 
{3} ! = (Pa-ers ar Pn—s+2 Fn sia) CPnoets + Dn 342) Pa=s41 
{s-1}! (CE eee Sey Nera) Yep oo eae ey eer) en Deen 
We see that for the set of elements p, p,-1. <5... - Dee 
the expression 
(1)n—s <2 (Oye 1 ne ret (Cee 
G@ {OH Cote ea iyt 
is 
1 1 ; 1 1 
@! Gmina Se antes 1) 
but for any set of s elements in any order 
1 1 ei 2 
one Gana sot RR RoR Bee Gay eS 
so that 
Be ee (2) Gea 
| ‘ (s)! Gas sree (1)! {s-1}! {s}!° 
and we have 
ve =. 8 Pn! 
ata SiGeniran 
2 ee ee! ey 
B, = 2 1) aoa )) 
which establishes the form of B, in general. 
(24) 
(25) 
(26) 
(27) 
