.MAJOR P. A. MACMAHON ON THE THEORY OE PARTITIONS OF NUMBERS. 377 
To adapt the function to enumerate the partition,s into at most m parts of /-partite 
numbers, such partitions being sucli as possess regular graphs in soJido, put 
X]i = Xj._, = Xi;j = . . . = — a I 
Xo] X'w X23 — . . . — Xoj,j — oi-i 
and the resultiim- function enumerates bv the coetEcients of 
“ O ^ 
rpVl/ytVi rp'pl 
tA^ I €A/'> • « • j 
the number of partitions of the /-partite 
into at most 7 U parts. 
Art. 94 , Further iDutting 
i O 
(7hP2 • . - it) 
. — .I'l — a‘, 
the coefficients of x" gives the number of graphs in aolido or unipartite partitions 
upon a tv/o-diniensional lattice, limited, as indicated above, by the numbers I, m, n. 
This function appears to be reducible to the product of factors shown in the 
tableau below :— 
(/i -f 1) 
-f 2) 
(/t -F 3) 
{^it -j- yyi) ^ 
(1) 
■ (2) ■ 
(3) • • ■ 
(w) 
(n + 2) 
{ii + 3) 
()i -F 4) 
(a -F m -F 1) . 
(3) 
(3) * 
(4) • • • 
(/;i + 1) 
(// A ;!) 
(// + 4) 
(/' + 0 ) 
(//. -F /// 4- 2) . 
(3) 
• (P ‘ 
(3) 
(7/r -F 2; 
(/t -j- /) 
(/t -f- ^ -j- 1) (/i -j- / -T 
2) (n “F HI -{■ 1 — 1 ] 
(/) ■ 
(1 -F 1) 
■ (1 + 2) 
{1 + — 1 ) 
This result, veritied in a multitude of particular cases, awaits demonstration. For 
/ = 2 it has been proved independently bv Professor Forsyth and l)y the present 
author. The diagTainmatic exhibition of the result at the jmiids of the lattice is 
clear, and since the product is an invariant for any substitution impressed upon the 
letters /, 7/1, n, it appears that such exhibition is six-fold. Taking a lattice whose 
sides contain I m poiiits respectively, so that I m points in all are involved, we mark 
a corner point, regarding it as an origin of rectangular axes one, and proceed to the 
opposite corner, along any line of route, such that progression along any branch or 
VOL. CXCIJ.—A. 0 C 
