MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 363 
For i — 2 , this may be shown to be equal to 
(1 — ic-'a+b (1 — 
(1 - (1 - 
(I 
— (1 ~ 
but for i > 2 , the functions are obtained with increasing labour, and are of increasing 
complexity. 
Many cases present themselves, similar to the one before us, where the H function 
is written down with facility, but no serviceable reduced function appears to exist. 
On the other hand, we meet with astonishing instances of compact reduced functions 
which involve valuable theorems. 
Art. 76. From the reduced function we can frequently proceed to an n function, 
thus inverting the usual process. If, for example, we require an fi equivalent to 
1 
1 - . 1 - .T- ... 1 - 
a little consideration leads us to 
n 
1 — . 1 
rt. 
1 a;P^-P, . 1 _ 
1 - 
('i-i 
This indicates that a unlpartite partition into the parts Pj, P 2 , . . . P^ may be 
represented by a twm-dimensional partition of another kind which involves the parts 
Po P 2 PlJ P.3 - P2> • • • Pi Pi-1' 
Ex. (jr., the numbers P,, Po, P 3 being in ascending order, the line partition 
P3P3P3P.2P.2Pj 
can be thrown into the plane partition 
Pi Pi P, Pi P, P. 
R-Pj P. 3 -Pj P. 2 -P. R-P, P,-Pi 
P3-P.2 P3-P2 P3-P2. 
of the nature of a regularised graph in the elements Pj, P .2 -- Pj, P 3 — P. 2 , though 
these quantities are not necessarily in any specified order of magnitude. We obtain, 
in fact, a mixed numerical and graphical representation of a partition of a new kind. 
If 
(Pj, R, P 3 ) = ( 1 , 3, 4), 
3 A 2 
