THEORY OF THE PARTITION OF NUMBERS. 
G41 
between the compositions and partitions of multipartite numbers. In the bipartite 
case we pass from any composition 
(Mi MM- -m) 
to the regularised partition 
S ih +i^~qT+q^ f 2h~+pT+~ p 3 > • • -m) 
of a certain bipartite number. 
The correspondence is between the compositions of p>q iuto 5 parts and the 
partitions of all bipartite numbers into s unrepeated biparts, the parts of the biparts 
being limited in magnitude to p and q respectively, and the highest bipart being pq. 
Or, we may strike out the highest bipart p>q, and then the partition is into s — 1 
unrepeated biparts, the parts of the biparts being limited as before. The partitions 
are subject to the further restriction that they are regularised in the sense that the 
unipartite partitions of p and q, that appear in the bipartition, are separately 
regularised. 
Art. 30. Instead of insisting upon this two-fold regularisation, we may, starting 
from the composition 
(/¥/n M* Pzfs> • • • M*)> 
proceed to the singly regularised partition 
(Mi lh+lh^h fiTb+ft, <h- ■ ■ Ms)- 
There are, in fact, various ways of forming connecting links between compositions 
and partitions of multipartite numbers whatever the order of multiplicity. These 
methods may be pursued at pleasure so as to obtain results of more or less interest. 
3. 
Art. 31. The correspondence set forth between unipartite partitions and bipartite 
compositions naturally suggests the possibility of a similar correspondence between 
bipartite partitions and tripartite compositions, and generally between m-partite 
partitions and m + 1-partite compositions. 
For the graph of the tripartite number pqr, we take r -f- 1 similar graphs of the 
bipartite pq, and place them similarly with corresponding lines parallel, and like 
points lying on straight lines ; the graph is completed by drawing these straight 
lines, which are in a new direction, say the y direction. 
There are three directions through each point of the graph (see loc. cit., Art. 31). 
There are 
V + 2 + r\* 
P> r l 
lines of route along which the tripartite compositions are 
* This notation explains 0 ■ 
pi q\ r I 
4 N 
and so in similar cases. 
MDCGCXCVI.-A. 
