372 MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 
Art. 87. I return to the enumerating function 
_ 1 _ 
(1 — a;) (1 — X")- (1 — x') 
to note that it may be exhibited as 
1 
12 7 7 ’ 
— 1 — ax . 1 - X . 1 -1- z 
a 0 c 
the interpretation of which is that the coefficient of x” in the development gives the 
number of instances in which 
“i + “2 "b “4 = '^1 > 
<^ 2 . “sj *4 being integers satisfying the conditions 
> a 2 >: S: a 4 . 
We arrive at the form in question if for these conditions we construct 
and then put Xi = Xa = X4 = x. 
The graphical representation is of the form 
1 1 1 1 1 1 1 I . . . 
1 1 1 1 I . . . 
0 0 0 0 ... 
11... 
the numbers of figures in the rows being in descending order and the third row of 
figures zeros. 
Art. 88. As another instance of the elementary lattice take the system 
^ a.,, > 1X3 
a4 > a., , a4 > a. , 
leading to 
1 
n-— 
- 1 - ahX, . 1 - 
ad 
1 - . 1 - cdX^ , 
reducing to 
__1 - XfXaX^X! _ 
1 - Xi. 1 - XiXoX^ . 1 - X 1 X 3 X,. 1 - X,X. 7 XsX 4 .1 - X 4 ’ 
