236 NOTE ON RIEMANN’S PAPER. [751 
of for some years) may perhaps be found interesting from its connexion with the 
theories of expansion and divergent series.” And I then give the expansion 
G n e x = [n — rf x n ~ 1 ~ r , 
where n is any integer or fractional number whatever, and the summation extends 
to all positive and negative integer values (zero included) of r. And I remark that, 
n being an integer, we have G n = T (n), and hence that assuming that this is so in 
general, or writing 
r (ft). e x — [ft — l] r x n ~ 1 ~ r , 
we have this equation as a definition of T (ft). The point of resemblance of course 
is that we have a doubly infinite expansion of e x in a series of integer or fractional 
powers of x, corresponding to Riemann’s like expansion of z x+h in powers of h. 
Cambridge, 10 Sept. 1879.