370 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
Art. 84 . We next have to observe the identity 
n _ 
” 1 — rt&Xj 
n _ 
> 1 - aXi 
1 _ 
1 — «&Xi X., 
1-X, 
a 
and to note that the dexter leads to the enumerating function 
2 : 1 — . 1 — ab.v^ 
corresponding to the problem of two superposable layers of units, each of two 
rows ; 
111111 1111 
1111 11 
in the case indicated superposition yields 
2 2 2 2 1 1 
2 2 11 
the first row contains a combined number of two’s and units > the combined 
numbers in the second row, and further, the number of two’s in first row, > the 
number of two’s in second row. In the 12 function these conditions are secured by 
the auxiliaries a, h, respectively, and it is established that the problem of partition at 
the points of the elementary [i.e., simple square) lattice is identical with that of two 
superposable unit-graphs, each of at most two rows. 
In fact, the graph 
2222 1 1.X 
2 2 11 
y 
the axis of 2 being perpendicular to the plane of the paper, is immediately convertible 
to the lattice form by projection, with summation of units, upon the plane y z. The 
numbers at the points of the square lattice would be 6, 4 , 4 , 2 respectively. 
