384 MAJOR P. A. MACMAHON OX THE THEORY OF PARTITIONS OF NUMBERS. 
further, the fact that the expression does give the enumeration when ^ = 1, for then 
the generating function is easily ascertainable to be 
1 -|- X ~j~ 3.r‘ -{“ 3x^ -b -}“ Oi'c'* ~t“ x' -b 
which may be exhibited in the forms 
(4y (5) ^ (3) (4y (5) _ (3) (3y^ (4y (5) 
a) (2y - (1) (2f (3) - (1 ) (2)-' (3)-" (4) ■ 
Art. 100. The second of these forms immediately arrests the attention, for, in 
piano, it denotes the number of partitions on a lattice of four points (in fact, a 
square), the part-magnitude being limited not to exceed 2. The reason of this is as 
follows :— 
Taking the cube with any distribution of units at the summits, we may project 
the summits upon the plane of y z, adding up the units on the cube edges at right 
X 
angles to that plane, and thus obtain a distribution, on the points of the cube face in 
that plane, of numbers limited in magnitude to 2. 
e- 
This projectton establishes the theorem, which may now be generalized. Conceive 
a lattice in solido having I, m, n points along the axes of x, y, z respectively, and a 
distribution of units at the points of the lattice which form an unbroken succession 
along each line of route through the lattice from the origin to the opposite corner, a 
line of route always proceeding parallel to the axes in a positive sense. Now project 
and sum unites on the plane of y z. 
