THE PARTITION OF NUMBERS. 
101 
For the unipartite case we put s = 1 and find, reading by rows, the numbers 
m s . 
to 2 , 
m. 
1. 
12 
15 
6 
1 
12 
18 
8 
1 
12 
21 
12 
2 
12 
24 
15 
3 
12 
27 
20 
5 
12 
30 
24 
6 
12 
33 
30 
9 
12 
36 
35 
11 
12 
39 
42 
15 
12 
42 
48 
18 
12 
45 
56 
23 
12 
48 
63 
27 
which admit of easy verification. 
In the notation of this paper, for the multipartite number 
m x m 2 ... m„ 
we have the enumerating number 
•MIIF (m f ; l 4 ) + 6 II F ? ; 1 2 2) + 3IIF2 2 ) + 8 II F ? (m,-; 13)+ 6 H F,(m < ; 4) 
Section II. 
Art. 16. The multipartite partitions which have been under consideration above 
have involved multipartite parts, and the integers which are constituents of those 
parts have been quite unrestricted in magnitude. We have now to consider the 
enumeration when these magnitudes are subject to various restrictions. 
The first restriction to come before us is that in which no integer constituent of a 
multipartite partition is to exceed unity. 
We form a fraction A l from Q ; by striking out from the latter every partition which 
involves a part greater than unity. 
Thus 
A t = l + (l) + (l 2 ) + (l 3 ) +.ad inf. 
We now form A,, A 3 , ... A„ ..., from A 1? by doubling, trebling, ... multiplying 
by i, ... all the bracket numbers of A 1} in the same way as we formed Q,, Q 3 , ... Q„ ... 
from Qi. 
