872 
MAJOR P. A. MACMAHON ON THE THEORY 
which is consistent also with the circumstance that it was found convenient analytically 
to regard the number of compositions of multipartite zero as being’ the fraction 
40. The number may be stated as the coefficient of {sa.)P[t^y[uyY in the expan¬ 
sion of 
1 _1_, 
2 1 — 2 (s« + + wy — tujB'y — usya. — stu^ + stua-lBy) ’ 
and we have shown that this fraction is equivalent to that portion of the expansion of 
the fraction 
1-- - - 
2 {1 — s(2a + /3 + y)} {1— ^5(2« + 2^ + y)} {t — U{2u + 2yS + 2y)} 
which is a function only of sa, and uy. 
41. It will now be shown that the fraction 
1_ 
Ml 
1 
Si (2«j + «2 + ... «„)} {1 — Sj (2«j + 2ix., + ... + «„)} ... {1 — s„ {2cii + 2«2 -F ... -f 2a„)} 
is, in fact, a generating function which enumerates the compositions of multipartite 
numbers of order n. 
This important theorem will be demonstrated by showing that the aggregate of 
terms in the expansion of the last written fraction, which is composed entirely of 
powers of is correctly represented by the fraction 
1 2 (Y ^ "t ■ ' * (. 1 ^1^2 * * ‘ 
which has been already shown to be a true generating function. 
42. For brevity put 
Sk = (2a^ -]- 2a2 + • • • + 2a^ + + i + . . . + a«) = (A,, + 2ctY) 
^ &0. ... 
M = (1 — (1 — . . . (i “ '28,^0.,^. 
The two fractions under comparison are 
\ 
2 
_ 1 _ 
— 1 + 2 (1 — S^a^) (1 — • • • (1 — S;.«h) 
1 
2N’ 
1 
2 
1 
(1 - Si)(l - So). ..(1 - S„) 
1 
2H ■ 
and 
