354 MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 
Similar results are obtained as solutions of linear Diophantine equations. 
The generating functions under view are real in the sense of Cayley and 
Sylvester. Enumerating generating functions of various kinds are obtained by- 
assigning equalities between the suffixed capitals 
X„ X„ . . . X,. 
Putting, e.g., 
X^j — ^^2 — • • • — — .r, 
we obtain the function which enumerates by the coefficient of x", in the ascending 
expansion, the numbers of solutions for which 
ct] “h cLo —j- . . , 0(3 n. 
It will be gathered that the note of the following investigation is the importation 
of the idea that the solution of any system of equations of the form 
“h A. 2 X 2 -{- -j- . . . -j- A-jOt, ^ 0 
(all the quantities involved being integers) is a problem of partition analysis, and that 
the theory proceeds pari passu with that of the linear Diophantine analysis. 
Section 5 . 
Art. 66. I propose to lead up to the general theory of partition analysis by con¬ 
sidering certain simple particular cases in full detail. 
Suppose we have a function 
F (x, a) 
which can be expanded in ascending powers of x. Such expansion being either 
finite or infinite, the coefficients of the various powers of x are functions of a which 
in general involve both positive and negative powers of a. We may reject ail terms 
containing negative powers of a and subsequently put a equal to unity. We thus 
arrive at a function of x only, which may be represented after Cayley (modified by 
the association with the symbol —) by 
fl F (x, a), 
the symbol > denoting that the terms retained are those in which the power of 
a is > 0. 
Similarly we may Indicate by the operation 
fi 
