NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 409 
(25.) Thus the expression for X becomes 
2<z^ + + s^x 
12. s 
x^^ + 2sxP + s^x^ 
247^ ? (*+«-!) 
(40.) 
6x^ + 1 ^x'^s + 1 Os'^x^ + s'^x 
720.S 
1)— &c. 
Y = - 1{ ’{'(4) - V ■ ’i"W ■ T"(s) '!""(*) + 
12 
5^+ 10s2-15s + 6 
720 
' 24 
&C. 
(41. 
(26.) We shall now proceed to the more immediate object of this paper, viz. the 
expression of the number of ways in which a given number x is susceptible of parti- 
tion, the number of parts being given. 
Let s be the number of parts into which x is to be divided, and let TI(7c) represent 
the number of ^-partitions of which it is susceptible. It is evident then that if 
A‘=l there is but one possible, so that in all cases '11(4')= 1. 
If s=2, the partitions stand thus : 
1,4—1; 2,4—2; 3,4 — 3; 
whose number is =. 
2 
Therefore we have 
X 1 ^ 
2 2 * 
If 5=3, the partitions so grouped as that none shall be twice repeated, will stand 
as follows ; — 
1, 1, 4 — 2 
2, 2, 4—4 
3, 3, 4 — 6, &c. 
1, 2, 4—3 
yo 
00 
1 
on 
&c. 
1, 3, 4 — 4 
&c. 
&c. 
The first column will consist of all the possible bipartitions of 4—1, each associated 
with 1, and their number is therefore ^11(4— 1). The second will consist of the 
bipartitions of 4 — 2, exclusive of (1,4—3), each associated with 2. Their number 
therefore will be identical with the total number of bipartitions of 4—4, because, so 
far as the number of cases is concerned, it matters not whether we consider 4—4 as 
parted into (1, 4—5), (2, 4—6), &c., or 4—2 as parted into (2, 4—4), (3, 4—5), &c., 
the reason of which will be obvious on trying any particular case. The number of 
terms therefore in the second column will be ^11(4 — 4). In like manner that in the 
third will be ^11(4—7), the bipartitions of 4—3 beginning with (3, 4—6), being 
3 G 
MDCCCL. 
