MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 365 
When the mixed elements 
Pj, P .3 P], P 3 Po, . . . 
are in descending order ot magnitnde we have a correspondence between nnipartite 
partitions and multipartite partitions of a certain class. 
Art. 78. It is usual to consider the parts of a partition arranged in descending 
order. The Xl function enables us to assign any desired order of magnitude between 
the successive parts. 
In the case of three parts we have already considered the system 
For the system 
we have the solution 
> a,, tto > oL-i. 
a, > a.,, a-^ > a.,, 
1 
“ 1 — fcX,. 1-X„. 1 — rt.,X» 
a^a., ' ■ ^ 
and thence the real reduced o-enerator 
1 - Xl . 1 - XiXoX,,. 1 - X;, 
and the enumeratinof function 
1 + .r 
(1. - ^’) (1 - (1 - 
On the other hand, for the system 
we construct 
^ , a, > a^, 
^ X X 
- 1 - ^.1 - ai«oX..l 
«■ 
n. 
leading to the real and enumerating functions 
1 - XiX|X» 
1 _ X.,. 1 - X,X,,. 1 - X.,Xo. 1 - X,X.,X3 
_ 1 + 
(1 - O (1 - (1 - '0 ’ 
of the former, the denominator shows the ground solutions, id est, fundamental 
partitions, 
(a„a„a3) = (0, 1,0); (110); (Oil); (111); 
and the enumerator points to the syzygy 
X,. X,X2X3 - X,X,. X.,X, = 0. 
