THE PARTITION OF NUMBERS. 
87 
and, regarding Qj as F(a, /3, y, ...), it is obvious that 
Q, = F (a‘, p,y\ ...); 
showing us that 
Q, = l+(2) + (4) + (2 3 ) + (6) + (42) + (2 3 ) + ... , 
Qt = 1 + (t) + (2 1 ) + (i 2 ) + (3*) + (2 i, 1 ) + (^' J ) + ... . 
Thence the expansion 
where 
1 4-aQ,! 
+(Qi 3 + 30A 2 +2Q 3 ) 
+ —. (On 4 + 6<« 2 + 3Q,," + 8 QjQ 3 + 6Q 4 ) 
4 ! 
+ . • • 
+ a*F ft (Q) 
+ . . . 
F, 
|*l2 /c 2 
^1, 
k \! k 2 
k ,! 
Art. 6. The importance of this expansion lies in the fact that the infinite series of 
Hammond operators on the one hand and the infinite series of Q, functions on the 
other hand have very remarkable properties in relation to one another. The first 
property we notice is that from the well-known law of operation, 
D m Qi — Qo 
for all values of m. Also 
E> m Q 2 = 0,2, or zero, 
ascending as m is, or is not, a multiple of two. And generally 
D m Q t = Q t , or zero, 
according as m is, or is not, a multiple of i. 
When is performed upon a product of k separate functions, it operates through 
the medium of a number of operators associated with the compositions ot m into k 
parts, zero being regarded as a part and D„ being regarded as a symbol for unity. 
Thus the compositions of the number 4 into three parts being 400, 040, 004 ; 310, 
