96 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
The investigation is therefore in six parts, 
(i.) Since 
T) 0 3 — 
6m + 2 
Qi'\ 
D 67 AQ 2 — (3m+l)Q 1 Q, 2 > 
D 6w Q :! = Q 3 , 
we find 
Demi (Qa 3 + 3 QiQ 2 + 2Q3) = i 
and immediately 
9 ) Qd+3 (3m+l) QxQ 2 +2Q 3 [•; 
D 6 mi I ) 6ni2 ... (Q|"+3Q,Q 2 +2Q;;) — 
1 !/ 6 W 1 + 2\[67n 2 + 2\ (6m s + 2\ q 3 
2 
6m + 2 
+ 3 (3m 1 + 1) (3m 2 + 1) ... (3m s + 1) Q,Q_, 
+ 2Q 3 |; 
and in particular 
(Qd+3 QiQ, 2 +2Q3) = i 
results which establish that the partitions of the multipartite number 
6m 1 6m 2 ... 6 m s , 
into three or fewer parts, are enumerated by 
Qf + 3 (3m +1 ) s QiQo + 2Q 3 r ; 
i ( 6 ”V 2 ) 6m ; +2 • ..( 6m i +2 )+ 3(Sm 1 + l)(3m,+ l)...(3m,+ l) + 2 
(ii.) Since 
we 
find 
D 6 m +1 Q 1 3 =( 6m 2 +3 )Q 1 3 , 
D 6 m+ iQaQi 2 = (3m +1) QiQo, 
D 6m+ iQ 3 = 0, 
7 6 m + 3 
Qr+3 (3m+ 1 ) QriQ>_. r; 
D 6m+ ii(Qi" + 3Q, 1 Q 2 + 2Q ;i ) — i 
and immediately 
D 6 » ltl D,.„. - i (Q, 3 +3Q,Q S +2Q 3 ) = i{( 6m y 3 ) ( 6m f 3 )... ( 6m | +3 )<» 1 8 
+ 3 (3mj +1) (3m 2 +1)... (3m s +1) QlQ.. 
establishing that the partitions of .the multipartite number 
6 m x + 1 6 m 2 + 1 ... 6 m, + 1 , 
