The portions (λ1λ2), (λ3λ4λ5), (λ6), ... are termed separates, and if λ1 + λ2 = p1, λ3 + λ4 + λ5 = q1, λ6 = r1... be in descending order of magnitude, the usual arrangement, the separation is said to have a species denoted by the partition (p1q1r1 ...) of the number n.
Definition II.—If in any distribution of n objects into n parcels (one object in each parcel), we write down a number ξ, whenever we observe ξ similar objects in similar parcels we will obtain a succession of numbers ξ1, ξ2, ξ3, ..., where (ξ1, ξ2, ξ3 ...) is some partition of n. The distribution is then said to have a specification denoted by the partition (ξ1ξ2ξ3 ...).
Now it is clear that P consists of an aggregate of terms, each of which, to a numerical factor près, is a separation of the partition (sσ11 sσ32 sσ33 ...) of species (p1q1r1 ...). Further, P is the distribution function of objects into parcels denoted by (p1q1r1 ...), subject to the restriction that the distributions have each of them the specification denoted by the partition (sσ11 sσ32 sσ33 ...) Employing a more general notation we may write
Xπ1p1 Xπ2p2 Xπ3p3 ... = ΣP xσ1s1 xσ2s2 xσ3s3 ...
and then P is the distribution function of objects into parcels (pπ11 pπ22 pπ33 ...), the distributions being such as to have the specification (sσ11 sσ22 sσ33 ...). Multiplying out P so as to exhibit it as a sum of monomials, we get a result—
Xπ1p1 Xπ2p2 Xπ3p3 ... = ΣΣθ (λl11 λl22 λl33 ...) xσ1s1 xσ2s2 xσ3s3 ...
indicating that for distributions of specification (sσ11 sσ32 sσ33 ...) there are θ ways of distributing n objects denoted by (λl11 λl22 λl33 ...) amongst n parcels denoted by (pπ11 pπ22 pπ33 ...), one object in each parcel. Now observe that as before we may interchange parcel and object, and that this operation leaves the specification of the distribution unchanged. Hence the number of distributions must be the same, and if
Xπ1p1 Xπ2p2 Xπ3p3 ... = ... + θ (λl11 λl22 λl33 ...) xσ1s1 xσ2s2 xσ3s3 ... + ...
then also
Xl1λ1 Xl2λ2 Xl3λ3 ... = ... + θ (pπ11 pπ22 pπ33 ...) xσ1s1 xσ2s2 xσ3s3 ... + ...