842 
MAJOR P. A. MACMAHON OX THE THEORY 
as was shown, loc. cit., a one-to-one correspondence between the compositions of the 
unipartite number m and the perfect partitions, comprising m parts, of the whole 
assemblage of unipartite numbers. 
Defining a perfect partition of a number to be one which contains one, and only 
one, partition of every lower number, it was shown that if 
(a, y, 8, , . 
be any composition of the number 
« + -f y + 8 + . . ., 
the partition, 
(H . (1 -f a)® . (1 + « . 1 + . (1 -f a . 1 +/3 . I +7)^ . 
the exponents being symbolic, denoting repetitions of parts, is a perfect partition of 
the number 
(l+«)(l+/3)(l+y)(L+8)...-l. 
§ 2. Multipartite Numbers. 
9. The multipartite number otySy . . . may be regarded as specifying « -j- /3 -j- y + . . , 
things, a. of one sort, of a second, y of a third, and so forth. 
To illustrate partitions and compositions of such numbers, I write down those 
appertaining to the bipartite number 21. 
Partitions. 
Compositions. 
(21) 
(^) 
(20 OT) 
oT), (dl 20) 
(11 To) 
(HTb), (Toll) 
(To^lTi) 
(Id^ dl), (Td dl Id), (dl Id^) 
I speak of the parts of the partition or composition, and observe that each part is 
a multipartite number of the same nature, or order of multiplicity, as the number 
partitioned. 
There is [see Art. 8, loc. cit.) a one-to-one correspondence between the compositions 
of the multipartite 
a/3y ... 
and the perfect partitions of the unipartite number 
... — 1 , 
wliere a, b, c, . . . are any different prime numbers. 
This correspondence is entirely distinct from that alluded to in Art. 12, 
