306 SiE William Rowan Hamilton on Fluctuating Functions. 



P4III + 4 „4 



^ fi?aN„a"">=\ c?aN„(a-f-4m)''' 



= Am log (4m) — (8m -|- 2) log (4m -|- 1 ) 

 + (8m + 6) log (4m + 3) - (4m + 4) log (4m + 4) 



But, by(h''), 



if A; be any integer number > ; therefore 



1 0~2t ,„^ 2* 



^ = 2, 



^""^(^ + i) 



/■7r\ 



ft»2;t being by (q'') the coefficient of ^-* ' in the development of tan x. From 

 this last property, we have 



^m -0^^ = t (S^ d^) t^" ^ = 1 S] ^^ log «ec a: ; (v^O 



therefore, substituting successively the values ^ = ^ and ^ = t, and subtracting 

 the result of the latter substitution from that of the former, we find, by (u'^^^), 



Q - - 



^ = - f Y^ — y j dr log sec s 

 = -y dx log tan s 



4 



8 r*- 



- y rf*- log cotan z. ( w ''") 



TT . 

 



Such, in the present question, is an expression for the constant w ; its numerical 

 value may be approximately calculated by multiplying the Napierian logarithm 

 of ten by the double of the average of the ordinary logarithms of the cotangents 

 of the middles of any large number of equal parts into which the first octant may 

 be divided ; thus, if we take the ninetieth part of the sum of the logarithms of 



