EA (nar...), (p.s„,.,.)  . (pqr...), where the numbers pi, qi, 1.1 . . . are fixed and assumed to be in descending order of magnitude, the summation being for every partition (pqr . . . ) of the number n, is defined to be the distribution function of the objects defined by (pqr . . . ) into the parcels defined by ( M i n .. . ) . It gives a complete enumeration of n objects of whatever species into parcels of the given species . 1 . One-to-One Distribution . Parcels m in number (i.e. m = n) .

Let h, be the homogeneous product-sum of degree s of the quantities a, ,B, y, . . . so that (1— ax . 1—13x . 1 --x . ...)—1=1 +hix -f-h2x2+1a3x3 + ... hi =

Ea = (1) h2 = Ea2+Ea'a = (2) +(12) h3 =Ea3+Ea2f+Eafly = (3) +(21) +(13) . Form the product hpinglhri .. . Any term in he, may be regarded as derived from pi objects distributed into pi similar parcels, one object in each parcel, since the order of occurrence of the letters a, 0, y, . in any term is immaterial . Moreover, every selection of pi letters from the letters in a5/39y' .. . will occur in some term of hy1, every further selection of q1 letters will occur in some term of h51, and so on . Therefore in the product h51h51hrl ... the term al'i32y' . . ., and there-fore also the symmetric function (pgr ...

), will occur as many times as it is possible to distribute objects defined by (pqr .) into parcels defined by (

Awl ...) one object in each parcel .