98 
MAJOR P. A. MacMAHON : SEVENTH MEMOIR ON 
the enumerating number 
i\r\ + 6 ) ( 6 , Y 6 )■ • • ( 6m ' 2 + 6 ) + 3 (3m, + 3) (3m a + 3)... (3m,+ 3) 
(vi.) Lastly, since 
rw,Qi 3 
bbm + sQnQ 2 
1^6171 + 5^3 
6ni + 7 | q 3 
(3 711 + 3) QiiQrj, 
0 , 
we obtain, for the multipartite number 
the enumerating number 
6 ?^ + 5 6 m 2 + 5 ... 6 m t + 5, 
l 176m,+ 7\ 
6 1 \ 2 ) 
( 6 T 7 ) 
( 6 Y 7 ) 
3 + 3) (3m 2 + 3) ... (3wij + 
Finally, in the notation employed for the bipartite case, for the multipartite 
number 
m 1 m 2 ... m, 
we have the enumerating number 
ijlIF 2 (m,; l 3 ) + 3 II F ? (m t ; 1,.2) + 2 II F ? (m t ; 3)j. 
Art. 14. I collect these results :— 
Multipartite Numbers. 
Number of Partitions into three or fewer parts. 
6 m 6m 6m repeated s times 
_1_ 
3! 
{( 6 T 2 )‘ 
+ 3 
(3m+l) s + 
6m + 1 6m + 1 6m + 1 repeated s times 
6m+ 2 6m+ 2 6m+ 2 ,, ,, 
6m+ 3 6?n + 3 6m + 3 ,, „ 
6m+ 4 6m+ 4 6m + 4 ,, „ 
6m+5 6m + 5 6m + 5 ,, „ 
1 f/ 6 m + 3V 
3!1\ 2 ) 
i{( 6 wi 2 + 5 )‘ + 3 ( 3m+2)- + 2 } 
|- ! {( 6 m 2 + 6 )'+3(3m + 3)'} 
i{( 6 “ + 7 )' + 3(3,„ + 3)]- 
