14 : THE REV. F. H. JACKSON ON 
When p=1 the right side of the differential equation vanishes and the whole equa- 
tion reduces to 
(2 -AHTE + e+) — (m=v)(ntv 4 Ly =0 
2. 
In this article we shall show that 
nm—vi||n n—-v|[n—v—1][n-v—-2][n-v-3 dy 
1- 7] Ip +p f ta ey ee... 7) 
of which the general term is 
pret [n-v|[m-v-l..... [n-v-2r+1_ pr 
[2] [4]... [2r]-[2n—-1]..... [Qa- or+1]y 
may be expressed as a convergent infinite product 
M((n])((m+r})-(2)n » & 
I([v])([2n]) - (2), 
reducing when 7 and » are integers to 
[7]! ! [me ab v]! (2)n 
(v]! (2n}i (2), OS ee 
as was anticipated in art. (10), Part I., by analogy with the sum of 
1 n-v-n—-v-l n—-v-n—-v—l-n-v—-2-n-v-3 
2-2n-1 Ke 2-4.I9n—-1-2n-3 
_ ni n+y! 2” 
yt Ql 
In the Proceedings of the London Mathematical Society, series 2, vol. i., ‘ Series 
connected with the Enumeration of Partitions,” the following theorem is obtained :— 
Pre [Pes SPP eye. | 10) 
r=] 
: ‘ pztm-l, ae (Paes ae 1) why & (PK, S 1) 
pis denoting L (Pet PPR Do. PPD 
[ a] deno ng ae (Pez + 1)(P*-"z 4+ 1) lawns (P24 1) 
Leal 
ey A (P(e Ty  . (pe =) 
{oa te Ug dea OP Pe ae 
o a 1 BET I ec Se ae naeses Pp" -1 
1a ” Dip 2D eps oo . pe 
perm—u pi +m—1 , om, poe 
” 1 Se es Re Ebin1ey Sie 8 
a eee 
PrP. i. 8) ume Ps 
Ula... - Us 
Yao ais te Ys 
being sets of s independent elements. 
