94 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
we write 
D„Q, ! = F, (m; l 3 ). Q, 3 
then 
D m Q 3 = F 9 (m ; 2). Q 2 , 
Qr' + Qo) = l 2 ). Q^ + F, (m,; 2).QJ, 
F> Mi D W2 HQi 2 + Ql-) = i {F^O,; l 2 ) F q (m 2 ; 1 2 ).Q, 2 + F q (m l ; 2) F q (m 2 ; 2). Q 2 }, 
F OT] D m2 ... D(Qr + Q 2 ) = \ fl F g (m ; ; 1"). Q, 2 + |- II F ? (m i ; 2). Q J( 
leading us to the number 
iilF.K; l1 + fnF,(m,; 2), 
as the enumerator of the partitions, into two or fewer parts, of the multipartite 
number 
{m{m 2 ... m s ). 
The reader will observe that the algebraic expressions of 
F ? (m 4 ; l 2 ) and F q (mi ; 2) 
will depend upon the parity of m,. 
The notation has been adopted so as to save a multiplicity of formulae in certain 
cases. 
This of course solves the question of the factorization into two or fewer factors of 
the composite integer 
P\ P'2 ■ • • Ps ■ 
We have, therefore, solved completely the question of enumerating the bipartitions 
of multipartite numbers. 
What has been done as a question in the theory of distribution may be stated as 
follows. We are given an assemblage of any numerical specification and two boxes 
which cannot be distinguished from one another. We have found the number of 
ways of distributing the objects between the boxes. The similar question when the 
boxes are distinguished from one another is simpler and connected with the 
compositions of multipartite numbers. 
Art. 12. At this point it may be appropriate to give a statement in regard to the 
nature of the solution given in this investigation. 
The enumeration of the partitions of a unipartite number m u into k or fewer parts, 
is formed as a linear function of certain numbers a x , b u c x ...; the linear function being 
Acq + P-by + I’Cy + ... 
where the numbers A, /m, v, depend only upon k. 
