MAJOR P. A. MACMAHON ON THE THEORY OF PAJITITIONS OF NUMBERS. 373 
establishing the fundamental solutions 
{a„a„a„a,) = {l, 0,0,0); (1, 1, 0, l) ; (1, 0, 1, 1) ; (1,1,1,],); (O, 0, 0, l) ; 
connected by the syzygy indicated by 
Xi. XiX3X3X4. X, - X1X0X4. X1X3X4 = 0. 
Art. 89. A more general generating function connected with the elementary 
lattice and descending orders is 
/ d 
1 - 
1 - ahX^ . 1 - — X,, 
a 
1 
where now aj, aj, a-, are restricted not to exceed ji, jo, j~^, respectively, and of 
course 
^ j-i 
IV IV 
^3 ^ j, 
are conditions. 
It should be remarked that we examine the case of bipartite partitions with regular 
graphs hy putting X 2 = Xj, X 4 = X 3 . 
Part-magnitude being unlimited, the reduced function is 
_ 1 - XfX. _ 
1 - Xi. 1 - Xb 1 - XiX,. 1 -^XiX.,. 1 - XiXi ’ 
and is real. 
Art. 90. Leaving the particular case, I pass on to consider the general theory of 
partitions at the points of a lattice in two dimensions. It can be shown immediately 
that it is coincident with the theory of those partitions of all multipartite numbers 
which can be represented by regular graphs in three dimensions. For consider the 
superposition of any number of unit graphs, adding into single numbers the units in 
the same vertical line. We obtain a scheme of numbers 
Ctii c/12 
®21 ®22 ^23 • 
CI31 <232 
^41 
y- 
X 
