Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 12)

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
	        
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