MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 3G1 
I 
1 1 1 — X ,1 — ry .1 — xz ’’ 
1 — ax .1 — — y. 1 — — 2 
a a 
n 
n 
1 
1 + xy 
] I — . 1 — xy- 
l-crx.l-^^y 
1 
1 1 — X .1 — oxy 
1 — ax. 1 — — y 
VL 
1 4 xy 4 xy- 
1 1 — a;. 1 — xir ’ 
1 — a-'x . 1 — y 
a 
1 
1 
1 1 — A’. 1 — of y ’ 
\ — ax . \ — —y 
ar 
o 
1 
1 4 o:z — xyz — -xyr- 
1 — d'X. 1 — ay .1 — 
(C 
n 
n 
1 — ,r. 1 — y .1 — yz. 1 — xz" ’ 
1 4 xy + xz -i- oyyz 
1 1 1 — as. 1 — xy’ . 1 — xz- ’ 
1 — a-x .1 — — y. 1 — — z 
a 
1 
I — ,xyin — .r:av — yuo 4 oyyziv 4 xyzvr 
i 1 — A’ . 1 — y . 1 — . 1 — xw. 1 — inv. 1 — zw ’ 
I — ax. 1 — ay. 1 — az . 1 — ~ to 
1 
2--r 
“ 1 — rta;. 1 — «//. 1 — — z .1 — 
a, a. 
1 — xyz — xyw — xyzio 4 xy'zw 4 x-yzu: 
1 — X .1 — y .1 — xz. 1 — . 1 —• yz. 1 — yto " 
Art, 73. I pass on to consider tlic partitions oi niimbei's into parts limited not to 
exceed i in magnitude. 
The n function is clearly 
o 
,i+i 
> 1 — ejXi 
/ a., 
\ a, 
1 - X., 
I 
1 - ( X.; 
a., 
1 - X.. 
a.. 
uij inf. 
In this form I have not succeeded in elfecting the reduction, hut if vvc put at once 
X, = X, = X, = . . . = 4, 
1 
the reduced form is 
VOL. cxcii.— A. 
i — X .1 — x^ .1 — x‘... 1 — a' 
3 A 
