360 
Proceedings of the Boy at Society 
from which two results by the regula falsi we derive a better 
approximation, viz., 
. 733 '75 -?33 
iA •019 + -01 Ui 
i.e 
-7388.... 
Continuing in this way, it is found that 
Lim n=oo (cos n a?) = -7390852, 
a result correct to at least the fifth place. 
This peculiar constant we should expect to find connected in 
some way with tt, and that such a connection exists is easily seen 
from the formula — 
cos 
4a? 2 \ 
svv ’ 
from which we have 
(x) 
-(*- 5 ) (>-£)(»- 
4<o 2 \ 
5W 
An explicit expression of the same relation is obtained from the 
identity 
7 r la? 3 1 ‘ 3 x b 
cos — ^ = 2 “ * “ 2 “3 ~~ 2T4'5 
,-i 
which gives 
7r 1 to 3 1 • 3 CO 5 
w=: 2 _t0 “2*'3~2 T 4’5“ 
whence 
, 0> 3 3 CO 5 
’ r = 4 " + 3 + I' 5" + 
the result of the u reversion ” of which is 
cos 00 # ~~ - 
7 r 
1 ( 
- 
7 r 
4 3 . 4\4/ .3.4. 5\4 
5 
