664 
MAJOR P. A. MACMAHON OX THE 
(i - Pi x ) (i -1’ x ) ( x - fr * 
i-^*) i--), 
Pl-2 j Pi-\ 
(1 - »,?.»») 
V m /V ih% 1 
Pi% 
1 zl^- 
Pi-2^1—2 
1 - 
pi-iqi-i 
(1 -p.q^x*) 1 - - 
ibA’s 3 
~s ] ( i _ M£i ^3 
‘ / \ JWs 
L — ^,3 \ / ! _ 
Pi-iqi-ii'i-v 
Pi-\qi-\ r <'i-\) 
0-P i?i 
. .. af') ^ 1 
j? 3 ga 
M 
.T 
1 — 
Ps9s ■ 
iAgz ■ 
a*” 
1 — 
\ Pi—2*21—2 • 
X" 
1 - 
Pi-iqi-i 
forming a rectangle of to rows and / columns, the letters y>, q, r, . . . , m in number, 
each occurring with l—l different suffixes. 
I say that forming a fraction with unit numerator, having the product of these 
factors for denominator, we obtain a generating function for the arrangements defined 
by ( l ; to ; co ). 
The number of layers is restricted to l (be., I or less), and the breadth to to (be., m 
or less), but the graphs are otherwise unrestricted. Reasoning of the same nature as 
that employed in the simple case of two layers, enables us readily to construct this 
function. The function is redundant, as we only require that portion of the expansion 
whose terms are altogether integral. In this portion we put the letters p, q, r, , . . 
all equal to unity, and thus arrive at the reduced generating function. 
I recall that the predicted result is the reciprocal of 
(i -*)(i 
X(1 
Professor Forsyth has not yet succeeded in obtaining this result from his powerful 
method of selective summation. 1 hear from him that he has verified it in numerous 
particular cases, but that, so far, he has not been able to surmount the algebraic 
difficulties presented by the general case. 
As regards the final result, the tableau of factors possesses row and column 
symmetry. 
Simple rotation of the graphs through a right angle in the plane xy establishes 
this intuitively. 
— t 3 ) (1 — t 3 ) ... (I — x m ~ 2 ) (1 — x m ~ l ) (1 — x' a ) 
— x z ) (1 — a? 3 ).( I — x ™ -1 ) (1 — x m ) (1 — x m+1 ) 
X (l — X 3 ).(1 — x ' n ) (1 — •P ii + 1 )(l — .T m+2 ) 
X . . . 
X (1 - X 1 ) (1 - X , + l ) . . . . •.• ... (1 - .T ?+ "'- 1 ). 
