852 
MA.JOR P. A. MACMAHON ON THE THEORY 
also 
F (413) _ 2F (421) -I- F (43) 
= /(4, 0) + (3 + 4 X l)/(4, 1) + (3 + 4 X 3 + 10 X l)/(4, 2) 
+ (1 + 4 X 3 + 10 X 3 + 20 X l)/(4, 3) 
+ (4 X 1 H- 10 X 3 + 20 X 3 + 35 X l)/(4, 4), 
verified by 
3408 — 3776 + 768 
= 0 F 7 X 1 -h 25 X 3 + 63 X 3 + 129 X 1, 
= 400. 
§ 3. The Graphical Representation of the Compositions of Bipartite Numbers. 
17. The graphical method, that has been era])loyed in the case of unipartite compo¬ 
sitions, can be extended so as to meet the cases of bipartite, tripartite, and multipartite 
numbers in general. For the present the bipartite case alone is under consideration. 
The graph of a bipartite number {x>q) is derived directly from the graphs of the 
unipartite numbers {p>)^ (l/)- 
Take 5' + 1 exactly similar graphs of the number p and place them parallel to one 
another, at equal distances apart, and so that their left hand extremities lie on a right 
line ; corresponding points of the q' + 1 graphs can then be joined by right lines and 
a reticulation will be formed which is the graph of the bipartite number 
We have ri/i a graph of the number p, and g' + 1 such graphs parallel to one 
another ; and AJ a graph of the number y, and y> + 1 such graphs parallel to one 
another. • 
The angle between AK and AJ\q immaterial. 
The points A, B, are the “ extremities ” or the “ initial ” and “ final ” points of 
the graph. 
The remaining intersections are termed the “points” of the graph. 
The lines of the graph have either the “ direction ” AK or the direction AJ. These 
will be called the a. and /3 directions respectively. Through each point of the graph 
pass lines in each of these directions. Each line is made up of segments, and we 
