358 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
in order to derive enumerating generating functions corresponding to certain problems. 
In the last-written general case, the quantities X, /x, ... -17 being given integers, put 
as a particular case, 
= Xo = .. . = X, = cc. 
The reduced function is 
1 
1 - . 1 _ ’ 
and herein the coefficients of t’", in the expansion, give the number of partitions 
«!, a.,, aj, . . . 
of all numbers which satisfy the equation 
-f 2/x . a2 d' . . . + 217 . = w, 
«!, a2) • • • being in descending order. 
For the three particular cases considered above this equation takes the forms 
Oil “b ^2 “b • • • “b ■“ ^^5 
Xl^i “b f^2^2 “b • • • "b ~ 
di = w, 
connected with the reduced generators, 
_ 1 _ 
1 — X . 1 — . 1 — .. 1 — x‘ 
_1__ 
\ ... 1 — x:''i+'^2+ ••• ’ 
1 
(1 - xY ’ 
respectively. 
Further, we may separate Xi, X^, ... X; in any manner into k sets and put those 
which are in the first set equal to Xi, those in the second equal to X2., and so on, and 
so reach an enumerating function involving k quantities, Xj, x^, X3, . . . x*. 
Ex. gr. Put 
Xj = X3 = X5 — ... — Xi, 
X2 = X4 = Xg = . . . = 072, 
and suppose i even. We obtain 
_^_ 
1 — X’l . 1 — X 1 .V 2 1 — . 1 — xja-l... 1 — ’ 
