620 
MAJOR P. A. MAC MAHON ON THE 
reference to the perfect partitions of unipartite numbers. # Firstly, there is a 
one-to-one correspondence between the compositions of the multipartite 
a/3y 
and the perfect partitions of the unipartite 
a u b $ c y ... — 1 ; 
a, b, c, . . . being any different primes. 
Secondly, there is 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. 
These bonds are interesting but, for present purposes, trivial. We require some 
correspondence concerning partitions which are not subject to the restriction of being 
'perfect. 
Art. 3. I first proceed to explain an important link connecting unipartite with 
multipartite partitions arising from the notion of the separation of the partition of a 
number (whether unipartite or multipartite) into separates ; a notion which leads to 
a theory of the separations of a partition of a number' 5 ' which was partially set forth 
in the series of Memoirs referred to in the foot-note. The theory of separations 
arises in a perfectly natural manner in the evolution of the theory of symmetrical 
algebra, and is, I venture to think, of considerable algebraical importance. Up to 
the present time the theory has been worked out as it was required for algebraical 
purposes; the various definitions and theorems are scattered about several Memoirs 
in a manner which is inconvenient for reference, and it therefore will be proper, while 
explaining the connection with compositions, to bring the salient features of the 
theory together under the eye of the reader. It suffices for the most part to deal 
with unipartite numbers. 
The Theory of Separations. 
Definitions. 
Art. 4. A number is partitioned into parts by writing down a set of positive 
numbers (it is convenient, but not necessary to assume positive parts, and occasion¬ 
ally to regard zero as a possible part) which, when added together, reproduce the 
original number. 
The constituent numbers, termed parts, are written in descending numerical order 
from left to right and are usually enclosed in a bracket (). This succession of 
* See also “ The Perfect Partitions of Numbers and the Compositions of Multipartite Numbers.” 
The Author, ‘Messenger of Mathematics,’ New Series, No. 235, November, 1890. 
