MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 357 
Art. 68. By choosing to sum the expression 
SXpX?... XT, 
every solution of the given conditions has been generated. The same result might 
have been achieved by other summations such as 
... xr, 
Xi, X 2 , . . . Xi being given positive integers, or as 
sxp“‘^X2^-“=.., X“i-r“'X“\ 
We, in fact, may take as indices of Xj, X. 2 , . . . X^ any given linear functions of 
aj, . . . aj, and form the corresponding generating function. 
For the two cases specified, the O functions are 
n 
^ 1 -uiXi*. 1 
^ Xf^^... 1 - — xr 
«i " «i-i 
1 - aiXi .1-^—M- — ^...1--^ 
«i Ai a, Aj Aj_i 
and the reduced functions 
1 _ xtu 1 - xt‘X^^... 1 - xkx^=... Xf 
1 - Xi. 1 - X 2 .1 - X 3 . . . 1 - X; 
respectively. 
Generally for the sum 
+ /11C2 + . .. -h . , . + /u-joa + . . . + rjia; 
the two functions are 
1 
D 
1 - aiXkX^^u . .Xf‘. 1 - ^XpXr. . .Xr. . . 1 - — X?^X3u..Xf 
and 
1 — Xt^Xa*... Xf‘. 1 — ... Xf+'*‘... 1 — ... Xf+-"'^’’' 
Art. 69. In any of these instances we have i quantities at disposal, viz. : 
^1) X 2 ,... Xjj 
