MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 385 
The result is a partition of the number at the points of a lattice in piano whose 
sides contain m and n points respectively, the part-magnitude being limited not to 
exceed 1. The descending order in this lattice is clearly from the origin to the 
opposite corner in the plane y z along each of its lines of route. 
The enumerating generating function is 
jl + 1 ) (i+1) 
(1) ■ (2) 
(/ + 2) {L + 3) 
(2) ■ (3) 
a + 3) (I_+_^ 
(3) • (4) 
(I + 3) (I 4- m) 
(3) (7«) 
(^ + 4) (I VI 
“(4j~ • • ■ • (//i + 1) 
(I + 5) (^ + '11^ + 2) 
(5) • • • • _|_ 2) 
(y -f- -f- 1) (I v 2i) {I 'ifi n) 
{n) ’ (% -f 1) ’ {n P 2) ' ’ ’ (vi + n) 
Each factor may be supposed at a point of the corresponding lattice ; if any point 
is the 5"' along a line of route the factor is 
(M- 5) 
(•5) 
The number of points at which we place 
(/ + s) 
(s) 
is equal to the coefficient of x" in the expansion of 
a: (1 -f , . . -h { \ x . -p x"~^) 
that is of 
If m, n be in ascending order and h, denote the 5**' figurate number of the second 
order, this coefficient is 
^0-n) 
the term -j- being omitted because s is at most m + n — 1. 
Hence the generating function may be wiitten 
. = m^+n- 11 ( / + g) 1 
*=i 1. (s) J 
Art. 101. It is now important to show the connexion between this result and the 
original lattice in solido. 
3 D 
VOL. CXCIJ.—A. 
