390 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
The next step is to construct an O function which shall express these conditions 
and lead practically to the desired summation. 
Art. 107. First take 5=2; there is but one condition 
and the function Ls 
>: ao, 
— 1 — cr^Xj. I 
1 
I - Xi. 1 - XiXo ’ 
and every term in the ascending expansion of this function is of the required form, 
and no other forms exist. The general term being 
Xp (XiX2)“'^ > a2, 
we may call Xj and XjX, the ground forms from which all other forms are derived. 
Art. 108. Next take 5 = 3. The conditions are 
«! + -^3 > 0.0 
«i 
OLo 
leading to the summation formula 
- - KiaoXi. 1 - X„. 1 
^ "3 j 
the auxiliaries an, determining the first, second and third conditions respectively. 
The function is equal to 
1 
1 
- 1 -«i«.,Xi.l - - Xo. 1 - —X„X 3 
a, ^ a.-, 
= a 
— 1 — «iXi -1-X 2 . 1 — rtiXiX2Xg 
= Q 
1 - Xj. 1 - X 1 X 2 X 3 .1 - XiX!Xs ' 1 - Xj. 1 - X 1 X 2 . 1 - XiX-Xj 
1 - XfXlXg 
1 - Xj. 1 - XjX.. 1 - X 1 X 2 X 3 . 1 - XjXsXs ’ 
representing the complete solution. 
