GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
127 
we have 
«!! — p-^, — _p3,. 
I I — Pi^’ 
and thence — Pi2'^'2 ~ P 12 — ^12 (suppose); introducing 
quantity •— 
<^12 — “l2y'l2> 
a.2i = 1 / 0^125 
an 
where ^^7 ^6 ^ certain function of the quantities 
Pv P2J Pl2’ ^]> % 5 
but, numerically, may not be either zero or infinity. 
The matricular relation is 
(X;^, Xg) — (o-jj, a^.2 ) (x^, x^) 
I %n %21 
(Pi> 
^ l/“l2’ P-2 
and the redundant form 
_1^_ 
{1 *’2 "I" ^ 2 * 3 )} 
of a singly infinite character. 
cTo = 0 ; 
'rP - (2" - 1 - cTa) = 1. 
Art. 30. The case n — 2,. 
The matrix being that connected with the determinant 
we have the following relations 
1 ! » 
and thence 
where 
— Pi’ %2 — P-2’ ^33 — 
^Gn %2 I — Pl2’ I ^l\’ I ~ Pl3> I ®235 ^33 I = 
I <^Gl> ^22’ <^33 I — 7*123 i 
*^12%1 — 7125 ^Is'^Gl — 7 i35 ^23^32 — 7235 
(7i25 7i35 723 ) — (7'ii^3 ~ 7^125 PxPd, ~ Pi?,’ P 2 P?, ~ P- 23 ) > 
undetermined 
PS’ 
P 23 ’ 
