826]
NOTE ON A PARTITION-SERIES.
219
and therefore
1.2.3.4 ... = 1 — (1 + x) x + (1 + of) x 5 — (1 + of) x 1 ' 2 + ...,
which is Euler’s theorem.
It might appear that the identities used in the proof would also, for this particular
value a = — 1, lead to interesting theorems; but this is found not to be the case:
we have
x 1
O’
of
1.2.3’
&c.,
but the expressions in terms of these quantities for the products 2.3.4
contain denominator factors, and are thus altogether without interest
example,
2.3.4
. = 1 +
— x 2 + of + of
r
(1 + of) x 12
1
+ &c.,
., 3.4..., &c.,
we have, for
which is, with scarcely a change of form, the expression obtained from that of the
original product 1.2.3.4 ..., by division by 1, = 1 — x. And similarly as regards the
products 3.4..., &c.
Cambridge, June, 1883.
28—2