THE PARTITION OF NUMBERS. 
105 
so that 
Dii (A, 2 + A 2 ) = . 2A! 2 , 
D a *(A 1 > + A J ) = KAf+A,), 
and 
DMm 2 +A 2 ) = i. 2 -A,*, 
results which show 
(i.) that the multipartite number 
111 ... s x times repeated, 
has 2 S|_1 partitions into two or fewer parts ; 
(ii.) that the number 
222 ... s 2 times repeated, 
has one partition into two or fewer parts ; 
(iii.) that the multipartite number 
222 ... So times, 111 ... s x times, 
has 2 Sl_1 partitions into two or fewer parts. 
Ex. gr. the multipartite number 2111 has the four partitions 
(1111 1000 ), (1011 1100 ), (1101 1010 ), (1110 1001 ). 
For the partitions, into three or fewer parts, we have 
D 3 e (A 1 3 +3A 1 A 2 +2A 3 ) = ^ (A 1 3 + 3A i Ao+2A 3 ), 
Do i (A 1 3 + 3A 1 Ao+ 2A 3 ) = £ (SA^+SAjAo), 
(A 1 3 + 3A 1 A, + 2A ! ) = £ (sA^+SAjAo), 
to which we may add for symmetry 
Do-6'(-A-i 3 + 3A 1 Ao + 2A 3 ) = g- (Aj 3 + 3AjAo + 2A 3 ); 
we gather that 
D 3 j £ (A, 3 + 3A,A., + 2A 3 ) = l (A 1 3 + 3A 1 A,, + 2 A 3 ), 
Vi'l (A 1 3 + 3A 1 A 2 + 2A 3 ) = } (3^A 1 8 + 3A 1 A 2 ), 
D s 1 , -HA 1 3 + 2A 1 A 2 + 2A 3 ) = * ^'Ad + SA^o), 
D 3 DoT)'i' l (A, 3 + SAjA. + 2A 3 ) = £ (3. s - +i A 1 3 + 3A 1 A 3 ) ; 
and it follows that the multipartite number 
333 ... s 3 times, 222 ... s 2 times, 111 ... s L times, 
has g-( 3 Sl+s, + 3 ) partitions, into three or fewer parts, of the nature we are considering. 
VOL. CCXVII.—A. Q 
