MAJOE P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
1 
375 
n 
1 ■ 1 
1 - ^ X31. 1 - 
. 1 - 
^ /e,x,3 . 1 
ftj 
^2 ^2 "y" 
lU 0~ -^22 • I 
h A 
0 , S, ' 
a.. 
yiXis 
h 72 V 
7 A _23 
b, 7i 
A Y, X 
Co 7., 
33 
to I factors to I factors to I factors 
. . to w factors 
. . to m factors 
. . to m factors 
&c. 
If the part-magnitude be limited to n, we must place as numerator in the function 
\n + l 
a. 
I — ( ajaiXii . 1 — ( — /?iXi 3 ) ... to m factors 
n-t -1 
n + \ 
71+1 
1 — 1^61 — Xoij . 1 — X 22 j . . . to m factors 
&c. 
to I factors 
to I factors 
and if we please we may reject all the numerator factors except 
1 - 
Art. 92. The existence of the three-dimensional graph shows that this function 
remains unaltered, when X^j is put equal to x, for every substitution impressed upon 
the numbers 
I, m, n, 
but there is a still more refined theorem of reciprocity connected with a more general 
generating function. 
Suppose that the number of layers which involve 1 , 2 , 3, &c. rows be restricted to 
^l3 ^2» 4? • • • j 
that the successive layers are restricted to Involve at most 
Wi, rru, TO 3 , . . . rows; 
and that the successive rows of the layers are restricted to contain at most 
Tij, 712 , . . . units. 
We have then the comprehensive fl function :— 
