100 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
We have 
D 2 m=(»+im 
D 2m+1 Q! 2 Q 2 = (m+l) (m + 2) QrQ 2 , 
D 2m Q 2 3 = (m +1) Q 2 2 , 
D 2 to+ 1 Q 2 = 
D 3 ot QiQ 3 = (m + l)Q 1 Q 3 , 
D 3m+ iQjQ:i = (m +1) Q 4 Q :t , 
D 3 m+ 2 QiQ 3 = + 1 ) QiQ 3 , 
D 4m Q 4 = Q 4 , 
D 4m + iQ 4 = + = ^4)71 + 3^4 = 
Utilising these results and taking as operators D s 12m , D s 12m+1 , ... D s 12m+11 in succession 
we find for partitions into four or fewer parts :— 
Multipartite Numbers. 
12 m 12 m 12 m repeated s times 
12 m + 112 m + 112 m + 1 repeated s times 
12 m + 2 12 m + 2 12 m +2 „ ,, 
12m + 3 12m + 3 12m+3 
12m+ 4 12m + 4 12m+ 4 „ „ 
12m+5 12m + 5 12m + 5 ,, „ 
12 m+ 6 12 m+ 6 12 m+ 6 ,, ,, 
12m + 7 12m + 7 12m + 7 ,, „ 
12 m+ 8 12 m+ 8 12 m+ 8 ,, ,, 
12m+ 9 12m+ 9 12m+ 9 „ ,, 
12 m +10 12 m +10 12 m +10 „ 
12 m +11 12 m + 11 12 m+ll ,, 
Number of Partitions into four or fewer parts. 
1 f/l2m + 3 \ s + 6 ( 6w+ i) 2 S+3 ( 6w+1 ) s + 8 ( 4m+1 ) J+6 
/6m + 2 
4! 1\ 3 
J_ J/l2m + 4 Xs 
4! l\ 3 
1 f/l2m+ 5 
+ 12 
V 2 
+ 8 (4m + l) 5 
4! IV 3 
1 f/12m + 6 
+ 6 (6m + 2) 2s + 3 (6m + 2) s + 8 (4m+l) s 
+ 12 
/ 6m+ 3 
V 2 
4! IV 3 
1_ f/12m + 7V 
4! IV 3 
J_ {^12m + 8\ s +]0 ^6m + 4 
1 f/12m + 9 
4! I 
1 f/l2m + 10\* /6m + 5 Xs 
41U 3 l+1 " 
+ 8 (4m + 2) s 
+ 6 ( 6 m + 3) 2s + 3 ( 6 m + 3) s + 8 (4m + 2) s + 6 
2 j +8(4m + 2) s 
+ 6 (6m + 4) 2s + 3 ( 6 m + 4) s +8 (4m + 3) s 
+ 8 (4m + 3) s 
h. I( 12W 3 + 11 J + G ( 6m + 5 ) & + 3 {6771 + 5) s + 8 (4m + 3) s + 6 
1 |/12m+12 V +12 /6m + 6 
3 
4! IV 
1 f/l2m+13\ 
4! 
1 I/12m+14 
\ 2 J +8(4m + 4) s 
2 y+ 6 (6m + 6) 2s +3 (6m + 6) s + 8 (4m + 4) s 
4! 
3 
+ 12 
6m + 7 V 
2 / 
+ 8 (4m + 4) s 
