DIAMETER AND CIRCUMFERENCE OE A CIRCLE. 



First, if we take o = — 7t, or 120°, then sin a = -~|3 and (4) gives a = 3. 



Witli these values (5) gives 



3 ,- V, 1.3 1 . 2 . 3 2 1 . 2 . 3 . 3 3 



( 6 ) w = tJ 3 ( 1+ Y^+ 7 ^ r ~^ + 



2.3 r 3.4.5 T 4.5.6.7 



■■) 



1 



Secondly, if we take a = -^- n, then we have sin u = 1, and (4) gives a = 2. 



With these values (5) gives 

 (7) ?--2(l+0+I 



1.2 



1.2.3 



ith 



!_ 1.2.3.4 \ 



3.5' 1.3.5.7 + 1.3.5.7.9 ''' 7 



Thirdly, if we take o = — n, then *m o = — Ijjj and (4 gives a = 1. W 



these values (5) gives 



... 3 ,_/. 1 1.2 1.2.3 1.2.3.4 \ 



(8) * =_ J 3 i 1 + 273 + 3T475 + 4.5.6.7 + 5.6.7.8.9 / 



Fourthly, if we take u = — n, then sin a = Ji~ and (4) gives a = 2 — /27 ^ n 



this case the value of a not being a rational integral number, the series is not con- 

 venient for computation directly as in the preceding cases. But we obtain a 

 series which' may be expressed in the following form : — 



(9) 

 in which 



Pi 1 P j_ 1 2 P j. 1 ■• 2 • 3 p 

 1 + O a + 3T475 3 + 4.5.6.7 



P 2 = 4J|-(2-j2-), 



and generally after the second 



P t = AP ( _ 2 — 2 P,_ 2 . 

 The first two being known, the remainder are easily found by the preceding relation, 

 and then with these (9) gives the value of n as follows : — 



.828427 x 



1 





= 



2.828427 



.656854 x 



1 

 2.3 





— 



.276142 



970562 x 



1.2 



3.4.5 





= 



32352 



.568540 x 



1.2.3 



4.5.6.7 





— 



4061 



.333036 x 



1.2.3.4 



5.6.7.8.9 





= 



529 



.125064 x 



1.2.3.4.5 

 6.7.8.9.10.11 





= 



70 



114184 v 



1.2.3.4.5.6 







10 





.8.9.10.11.12. 



13 



71 









3.14159 



