THEORY OP THE PARTITION OF NUMBERS. 
665 
We get the same result from m rows and l columns as from l rows and m columns. 
Taking only the first row, we find that the fraction 
(1 
leads to the same reduced generating function as the fraction 
_1_ 
(! -pjx) (1 - p&a?) • • • (1 ~V\<h • • - xl ) ’ 
and this is obviously 
1 
(1 - x) (1 - x”~) ... (1 - X 1 )' 
Art. 59. If the result predicted be the true result we should be able to establish it 
by means of a one-to-one correspondence between the graphs and partitions of a 
certain kind. This presents difficulties to which I will advert in a moment. 
Finally I construct the generating function for the case ( l;m; n), the graphs being 
restricted in all three dimensions. 
The numerator is unity and the denominator the product of the factors exhibited 
in the subjoined tableau :— 
(1 -gp,q 1 x 1 )( 1 
.(i 
Vl-2 ( ll-2 ) \ Pl-1%1-1 
J5 
(1 - 9Pi<h ■ • • xm ) 
P&z • • • x m\ 
lh<h • • • ^ / 
Pi-iqi-i ■ • • 
'pi — 2 qi—2 . • • 
X m \ 
Pi-iSi-i • • • / 
in which the occurrence of the symbol g in the first column will be noticed. 
We have as usual to neglect all terms in the expansion which involve negative 
powers of symbols and in addition we must now neglect all terms which involve g 
raised to a greater power than n. 
This construction prevents the lower layer of the graph from having a greater 
extent than n in the direction 0 y, and thus the whole graph is similarly restricted. 
The reduced generating functions can be shown in simple instances to agree with 
the predicted results. 
MDCCCXCVI.— A. 4 Q 
