3(38 ]»1AJ011 P. A. MACMAIK'IN ()X THP THK(JRY Ob’ PARTITIONS OF NUMBERS. 
Section 6. 
Art. 81. The theory, so far, has been concerned with partitions upon a line. The 
parts were supposed 
a., ctj ^4 aj , . . a__, a, 
• -- • _ •-•- • 
to be placed at the points upon a line with one of the symbols < placed between 
every pair of consecutive points. 
When the symbol was invariably > the enumerating’ function found was 
(.y_+ 1) (j + -) {j + 3) ^ ij + 0 
Id ’ (D ' (3) (tr 
wherein [s) denotes 1 — x\ If we place these factors at the successive points of the 
line we obtain a diagrammatic exhibition of the generating function, viz. :— 
b/' + 3) (j + o) (y + T) (j + i — 1) D/+_y) 
"(D (2) (3) (4) ■■■ D-1) ~(i) 
a simple fact that the following’ investigation shows to be fundamental in idea. 
Art. 82. I pass on to consider partitions into parts placed at the points of a two- 
dimensional lattice. 
For clearness take the elementary case ol four parts jilaced at the points of a 
square. 
with symbols ^ placed as shown. We have to solve the conditional relations 
0L^ ^ ao, (X., 2: 
> as, as > a4. 
The four parts are subject to two descending orders. For the sum 
Ave have the f) function 
—! .lVj ..Vs ..Vs ..a4 
