MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 355 
that the only terms retained are those in which a occurs to the power zero and the 
meaning of the operations 
n, n, n, n 
^ > 
will be understood without further explamation. To generalise the notion we may 
consider 
n F ^^2? »• • ^ij ^2? • • • 
to mean that the function is to be expanded in ascending powers of Xj, Xj, . . . X„ 
the terms involving any negative powers of ai, a. 2 , . . . at are to be rejected, and that 
subsequently we are to put 
— 0^2 ““ • • • — — 1. 
In this case the operation n has reference to each of the letters ai, . . . at and a 
term involving any negative power of either of these quantities is rejected. 
If the quantities a^, . . . at be not all subjected to the same operation we may 
denote the whole operation by 
Ctj CLb 
n D. n,.... D. F (Xi, X2, . . . X„ «!, 02, Os . . • at) 
Or 
wherein operates upon .a^ according to the law of the symbol 0 -^. 
CTf 
The operation, qud a single quantity and the symbol >, have been studied by 
Cayley.* Qud more than one quantity it has presented itself in a memoir on 
partitions by the present author.! 
These fl functions are of moment in all questions of partition and linear 
Diophantine analysis. 
Art. 67. I will construct D. functions to serve as generators of well-known 
solutions and enumerations in the theory of unipartite partition. 
Frohlem I. To determine the number of partitions of w into i or fewer parts. 
Graphically considered we have i rows of nodes 
. 
0-2 ... . 
0.3 .. . 
a,- 
* “ On an Algebraical Operation,” ‘ Collected Papers,’ vol. 9, p. 537. 
t “ Memoir on the Theory of the Partitions of Numbers,” Part I., ‘ Phil. Trans.,’ A, vol. 187, 
pp. 619-673, 1896. 
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