182 
which will give, by (3), and by the principle last mentioned 
respecting odd exponents, 
(2X + Ls 2, 2v) ={A, ps Dis 
(2A - 1, Quy 2v) ={A- 1, [Ms v}. 
We shall then have, by the mere notation, 
DTN, Te PE” 5 (6) 
and, by treating this equation on the plan of (3), 
{Ay ps v}=(A-I, pm, v}+{A,u—-1,v}4+{A,m,v-1}. (7) 
By a precisely similar reasoning, attending only to j and &, or 
6) 
making \ = 0, we have an expression of the form, 
3 fr = (us v) pk, 8) 
where the coefficients {y, v} must satisfy the analogous equa- 
tion in differences, 
(mw vj=(u—-1, v}+{u,v—-1}, (9) 
together with the initial conditions, 
(us O}=1, (0, v}=1. (10) 
Hence, it is easy to infer that 
Ges oe 
1 es Se a”) 
one way of obtaining which result is, to observe that the ge- 
nerating function has the form, 
= {u, vj ute’ = (1 -u-v)t. (12) 
In like manner, if we combine the equation in differences (7), 
with the initial conditions derived from the foregoing solution 
of a less complex problem, namely, with 
(0, p,v}=(u vj}, {A, 0, v}=[A, v}, {A, ws OJ=[A, nw}, (13) 
when the second members are interpreted as in (11), we find 
that the (slightly) more complex generating function sought is, 
Dr, wy vj Puro = (1-t-u-v)7; (14) 
and therefore that the required form of the coefficient is, 
