G34 
MAJOR P. A. MACMAHON ON THE 
To obtain a general formula write 
_ (1 — oxp)p +1 
1 — a . 1 — x . 1 — ax . 1 — ax 3 ... 1 — ax p . 
(1 — axP~ l )P 
~ 1-a.I - x. I - ax.I - ax ?... 1 - carJ~ x ^ p ' X ' ’ 
then 
(1 — axP)P + l — (1 — axP) (1 — axP~ l )P 
1 — a . 1 — x. 1 — ax .. 1 — ax 3 ... 1 — axP 
axP - 1 {(1 - ffq??)p- 1 + (1 - a.xP)P-~ (1 - rt.i-?- 1 ) + ... + (1 — axP- l )P~ 1 } 
1 — a . 1 — ax. 1 — ax -... 1 — ax p ~ l 
and w r e have in succession 
u 0 (x) = y~i 
X 
U, (x) 
Uo (x) 
a 
1 — a 
ax , ax (1 — ax 9 ) 
_L_ 
1 — a 1 — a. 1 — ax 
T -r f x ax 9 (1 — ax") , ax" (1 — ax 3 ) | ax" (1 — arc 3 ) 2 
U 3 ) = i ~ ZZ + i Z “ + “ 
1 — a.l — ax 1 — a .1 — ax 1 — a.l — ax.l — ax" 
TT , . ax? (1 — ax')" a* 3 (1 — ax?) (1 — ax 4 ) 
U 4 (x) = -----„ + ----- - 
' 1 — a.l — ax. 1 — ax- 1 — a.l — ao:.l — ax- 
t ax s (1 — ax 4 )" , 
+ i „ -r ZZ Z + 
ax 3 (1 — ax 4 ) 5 
1 — a.l — ax. 1 — ax 9, 1 — a.l — ax.l — ax 2 .1 — ax 3 
and, in general, 
Up ( x) ^ 
axP 1 (1 — axP~ l )p- 
a .1 — ax ... 1 — axP 
~r> + 
axP 1 (1 — axP x ) p 3 (1 — axP) 
1 — a .1 — ax... 1 — axP~" 
axP 1 (1 — axP ^)r~ 4 (1 — axP ) 3 
1 — a.l — ax... 1 — axP~" 
axP 1 (1 — axP) v 3 
+ i z—I- 1 —Z + 
axP~ 4 (1 — axP)P~ l 
Hence, 
1 — a. 1 — ax ... 1 — axP 3 1—a.l—ax...1 — axP ~ 2 .1 — ax*~ l 
1 1 p 
_ _ _ —-1- vTT ( x \ . 
1 — x .1 — a .1 — ax .1 — ax "-... 1 — ax* (1 — axv)p +4 o p ' ’ 
a valuable expansion. 
Simple inspection of this formula shows that (1 — ax?)~ p ~ l represents that portion 
of the G.F. which is a function of ax p only. 
