794] 
NUMBERS. 
591 
We have such analytical formulae as 
1 _ zx z^x 1 
1 — xz .1 — x 9 z. 1 — a?z ... ^ 1 — x + \ — x .1 — 
which lead to theorems in the Partition of Numbers. A remarkable theorem is 
1 — x . 1 — x 2 .1 — a?. 1 — ¿c 4 ... = 1 — x — x 9 + oc? + x 7 — x 12 — ¿c 15 + . •., 
where the only terms are those with an exponent (3n 2 + n), and for each such pair 
of terms the coefficient is (—) n 1. The formula shows that, except for numbers of the 
form £ (3n 9 ± n), the number of partitions without repetitions into an odd number of 
parts is equal to the number of partitions without repetitions into an even number 
of parts, whereas for the excepted numbers these numbers differ by unity. Thus for 
the number 11, which is not an excepted number, the two sets of partitions are 
11, 821, 731, 641, 632, 542, 
10.1,92, 83, 74, 65, 5321, 
in each set 6. 
We have 
1— x.1 + x.1 + x 9 .1 + a?.1 + ofi ... = 1\ 
or, as this may be written, 
1 + #.1+# 2 .1+# 4 .1 + ¿c 8 ... = 
1 
1-x’ 
= 1 + X + X 2 + X 3 + ..., 
showing that a number n can always be made up, and in one way only, with the 
parts 1, 2, 4, 8,.... The product on the left-hand side may be taken to k terms 
only: thus if k = 4, we have 
] — x 16 
1 + x. 1 + x 9 .1 +« 4 .1 + ot?, = ^ , = 1 4- x + x 2 + ... + a? 5 , 
1 — x 
that is, any number from 1 to 15 can be made up, and in one way only, with the 
parts 1, 2, 4, 8; and similarly any number from 1 to 2 k - 1 can be made up, and in 
one way only, with the parts 1, 2, 4, ..., 2 k ~ l . A like formula is 
1 _ a? 1 — x 9 1 — x w 1 — ic 81 _ 1 — a? 1 ' 
x . 1 — x ' X s . 1 — a? ' x 9 .1 — x 9 ’ x w . 1 — ¿c 27 ic 40 .1 — x ’ 
that is, 
X- 1 + 1 + X . X~ 3 + 1 + X 3 . x~ 9 + 1 + X 9 . X~ 27 + 1 + X s = X~ i0 + X~ 39 + ... + 1 + X + ... + X? 9 + x*°, 
showing that any number from — 40 to + 40 can be made up, and that in one way only, 
with the parts 1, 3, 9, 27 taken positively or negatively ; and so in general any number 
from — |-(3 fc —1) to +\(2> k — 1) can be made up, and that in one way only, with the 
parts 1, 3, 9,..., 3* _1 taken positively or negatively.