lem of distribution, of which the partition of a number is a
particular case. He introduced the method of symmetric functions and the
method of differential operators, applying both methods to the two
important subdivisions, the theory of composition and the theory of
partition. He introduced the notion of the separation of a partition,
and extended all the results so as to include multipartite as well as
unipartite numbers. He showed how to introduce zero and negative
numbers, unipartite and multipartite, into the general theory; he
extended Sylvester's graphical method to three dimensions; and finally,
1898, he invented the "Partition Analysis" and applied it to the
solution of novel questions in arithmetic and algebra. An important
paper by G. B. Mathews, which reduces the problem of compound partition
to that of simple partition, should also be noticed. This is the problem
which was known to Euler and his contemporaries as "The Problem of the
Virgins," or "the Rule of Ceres"; it is only now, nearly 200 years
later, that it has been solved.

Fundamental problem.

The most important problem of combinatorial analysis is connected with
the distribution of objects into classes. A number n may be regarded as
enumerating n similar objects; it is then said to be unipartite. On the
other hand, if the objects be not all similar they cannot be effectively
enumerated by a single integer; we require a succession of integers. If
the objects be p in number of one kind, q of a second kind, r of a
third, &c., the enumeration is given by the succession pqr... which is
termed a multipartite number, and written,

______
pqr...,

where p + q + r + ... = n. If the order of magnitude of the numbers p,
q, r, ... is immaterial, it is usual to write them in descending order
of magnitude, and the succession may then be termed a partition of the
number n, and is written (pqr...). The succession of integers thus has a
twofold signification: (i.) as a multipartite number it may enumerate
objects of different kinds; (ii.) it may be viewed as a partitionment
into separate parts of a unipartite number. We may say either that the
objects are represented by the multipartite number

______
pqr...,

or that they are defined by the partition (pqr...) of the unipartite
number n. Similarly the classes into which they are distributed may be m
in number all similar; or they may be p1 of one kind, q1 of a second, r1
of a third, &c., wher