THE PARTITION OF NUMBERS. 
99 
Multipartite Numbers. 
6m 6m 6m repeated s times 
6m +1 6m + 1 6m +1 repeated s times 
6m + 2 6m + 2 6m + 2 ,, ,, 
6m T 3 6m "P 3 6m T 3 ,, ,, 
6m+ 4 6m + 4 6m+ 4 ,, ,, 
6m-f 5 6m + 5 6m+5 ,, „ 
Number of Partitions into exactly three parts. 
h {( 6? 2 + 2 J + 3 (3«» +1 )• -3 (6m+1 )*-1} 
(( 6?> 2 + 3 ) + 3 (3m+l)*-3 (6m + 2)*j 
J-j {( 6W 2 +4 )* + 3 (3m+ 2)*— 3 (6m + 3) s -3j 
|y j( 6W 2 + 5 )' + 3 ( 3 ™ + 2)*-3 (6m + 4)* + 2 } 
h {( 6W 2 + 6 J + 3 ( 3m + 3)'-3 (6m+ 5)*—3 j 
|j{( 6m 2 +7 ) S + 3 (3m + 3)*-3 (6m + 6)*|- 
As a verification, connected with unipartite partitions, we put s = 1 in these last six 
formulae, and reach the six numbers 
3 m 2 , 3m 3 + m, 3m 2 + 2m, 3m 2 + 3m+l, 3m 2 +4m+l, 3m 2 + 5m+ 2, 
and since these may be exhibited in the forms 
(6m) 2 (6m+1) 2 r (6m+ 2) 2 x (6m+ 3) 2 x (6m+ 4) 2 x (6m + 5) 2 ■, 
12 ’ 12 TZ> 12 12 12 3 ’ 12 
we verify the well-known theorem which states that the number of tripartitions of n 
is the nearest integer to — . 
12 
The Partitions of Multipartite Numbers into Four Parts. 
Art. 15. The operand is 
ft (Qd+6Q, 2 Q,+3Qy + 8Q]Q 3 +6Q 4 ) 
since the result depends upon the divisibility of m by the numbers 2, 3 and 4, and 
12 is the least common multiple of those numbers, it will be necessary to take the 
operator suffix to the modulus 12, and the investigation is, therefore, in twelve 
parts. 
p 2 
