MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 371 
Ai't. 85. Observe too tliat the partition is also one upon another kind of lattice in 
which the part-magnitude is limited not to exceed 2. 
2 2 2 B ! I 
...... . BS 
^.211 
y 
Here, starting from the origin, we may proceed to the opposite point of the lattice 
along any line of route which proceeds in the positive direction along either axis, and 
the condition is that along each line of route (here there are six) the numbers must 
be in descending order and limited in magnitude to 2. 
Art. 86. We have, therefore, solved the system of conditions : 
aj > a., ^ ^. — 
IV IV IV IV 
> yds >.> 
2 > > 0 , 
which is seen to possess the same solution as the system 
*3 ^ “4 
y > > 0 ; 
and we remark the diagrammatic representation 
the product of all the factors being 
(j + 1) (j + 2)-- (j + 3) 
(1) (2)^ (3) 
3 B 2 
