374 MAJOR P. A. MACMAHON OR THE THEORY OF PARTITIORS OP RCJMBERS. 
in which all the rows and all the columns taken in the positive directions along the 
axes of X and y are in descending order. We may consider these numbers to be 
placed at the points of a lattice of which the sides involve in and I points along the 
sides parallel to the axes of x and y respectively ; m will then be a limit to the 
number of units in any row of a unit graph, and I will be the limit to the number of 
rows. 
- s -' 
> ■ ' ' i 
w 
--- 
w 
• 
- E - 
y 
There is a descending order along each line of route from the origin to the opposite 
corner of the lattice, and there are altogether 
7 + m — 2 \ 
, ^-1 J 
such lines of route. 
Art, 91, The theory of the regular partitions of multipartite numbers is thus 
reduced to a lattice partition into I m parts in 'piano. The conditional relations may 
be written 
an — “i2 ^ “13 
IV IV IV 
a 
1. m—1 
> a 
Ini 
IV IV 
aoi > a.,2 — a.j3 
a, 
2 . 7n — 1 
— 1 — Ot .9 
— “/-1.2 — ®Z-1.3 
IV 
IV 
IV 
a 
i—1 .m —1 
a. 
2 — 1 . m 
IV 
IV 
a 
2,1 
> aj.2 ^ ai.3 
aom-i — a. 
I . TK 
and for the sum 
S n n X 
dti 
it 
i-\ t=\ 
we at once write down the H generating function, viz, 
