MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUAIBERS. 391 
The denominator factors yield the ground forms 
X1X2X3, X,X 1 X 3 
in addition to those previously met with, whilst the numerator factor indicates the 
ground form syzygy 
X,. XiX;X 3 - X1X2. X1X3X3 = 0. 
Observe that 
XJX 2 X 3 — 
X 1 XIX 3 = 
1 - x”+^. 1 - a;”+- ■ 1 - 
1 — a:. 1 — 53^. 1 — 
(1 - x”+^) (1 - x’^+^-y (1 - a;”+“) 
(1 - X) (I - x^y (1 - x^) 
are those with which we are familiar in the theories of simple and compound 
partition respectively. 
Art. 109. I pass on to the case 5 = 4; the conditions are 
"T *^3 — ^-2 “h 
> a3 
a., > ag 
Oil > 
OL, > 
ag > a.^ 
We neglect the fifth of these as being implied by the remainder and from the 
function 
n 1 
1 — .1-^ X 2 . 1 
a,, , aiCig -^2 ^'^2 -t- 
a.,a.. 
which, when reduced, is 
+ 
+ 
1 - Xi. 1 - X 1 X 2 • 1 - XjX^XsXj. 1 - XjXlX^X, 
_XXIX3_ 
1 - Xi. 1 - X1X2. 1 - X1XIX3. 1 - XiX|X:3X, 
_X1X2X3_ 
1 - Xi . 1 - X1X2X3. 1 - X1XIX3 . 1 - XiX|X|X, 
showing that the new ground forms are X 1 X 2 X 3 X 4 and XiX 2 X 3 X 4 , both of which have 
presented themselves before. 
The result may be written 
1 - X?X|X3 - XfX|XlX4 - XfXlX^Xi + X?X|X2X4 + X?X^X|X4 
1 - Xi . 1 - XjXg . 1 - XiXjXs . 1 - X1XIX3 . 1 - X1X2XSX4. 1 - XiXIX^X, 
