599] 
a smith’s prize dissertation. 
261 
The formula shows, not only that B n is a rational fraction but that its denominator 
is at most = least common multiple of the numbers 2, 3, ..., 2w + 1 ; the actual 
denominator of the fraction in its least terms is, however, much less than this, there 
being as to its value a theorem known as Staudt’s theorem. It does not obviously show 
that the Numbers are positive, or afford any indication of the rate of increase of the 
successive terms of the series. 
These last requirements are satisfied by an expression for B n as the sum of an 
infinite numerical series, which expression is obtained by means of the function cot0, 
as follows : 
We have 
or, writing herein t = 2iÔ [i = V(— 1) as usual}, this is 
6 cot 6 = 1 — Bj y~2 ~ 
But we have 
and thence, by differentiation, 
— «& c. 
Hence 
that is, 
showing first, that B n is positive, and next, that it rapidly increases with n, viz. n being 
large, we have 
2(1.2... 2n) 
Bn ~ (2ir)™ 
or, instead of 1.2... 2 n writing its approximate value V(2tt) . (2n) m+ h~ m , this is 
£ w = 4V(mr) ^ .