THEORY OF THE PARTITION OF NUMBERS. 
G51 
in a subjacent succession of lines in the bipartite graph. This species of regularisa- 
tion is the natural extension to three dimensions of Sylvester’s graphical method in 
two dimensions. 
Art. 43. Sylvester represents the partition (ctp^ag...) of a unipartite number A 
by the graph 
9 ® © © 
® © © © 
® 9 © 
the lines containing a x , a 2 . « 3 . .. nodes successively. 
The same graph also represents a multipartite number (ci x (L 2 a z . ..) whose content 
is A, viz., 
a \ A a -2 + a % + • • • — A. 
Sylvester’s theory is, in fact, not only a theory of the partitions of a number A, 
but also a theory of the multipartite numbers whose content is A. For purpose of 
generalization I prefer to regard it from the latter point of view. 
If we consider the graphically regularised partition 
^CL-yCL C)Cb Q • . • , * • • J C^CoCg ..., .| 
of the multipartite number 
(a 1 + by -f- Cj + . . ., a 2 + b 2 + c 2 + . . ., a 3 + b 3 + c 3 + . . .,.) 
and write down the Sylvester-graphs of the multipartite numbers which are the parts 
of the partition 
A 
© O <5 • • © 
© © O • 
• • tt 
3 
© © © a • • 
• ® o 
© © 
c 
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it is clear that we may pile B upon A, and then C upon B, &c., and thus form a 
three-dimensional graph of the partition 
0 \ --3T 
(®) (§)(§)(®) ® ® 
(®) ® ® # 
(§) ® • 
y 
which is regularised in three-dimensions just as the Sylvester-graphs are regularised 
in two. 
4 o 2 
