6 DIAMETER AND CIRCUMFERENCE OF A CIRCLE. 



of a, converge slowly. In (8) in which a = 1, the limiting ratio of convergency is f. 

 In (9) in which a = 2-/2 the limiting ratio of convergency is £ (2— J%)= } 

 nearly. In (10) in which a = 2 — J3 the limiting ratio is J (2 — J3 ) = T \ nearly. 

 This is much more convergent than the formulae of Clausen and Dase of Germany, 

 with which they separately and independently computed the value of n to 200 

 places. It has also the advantage over their formulae in giving the value of n from 

 the computation of one series only. It is, however, less convergent than the for- 

 mula of Machin, in which the limiting ratio, in the less convergent of the two series, 

 is ^ ; but Machin's formula also requires the computation of two series. 



All of the preceding series are very interesting on account of the extremely simple 

 laws of their continuance. 



PUBLISHED BY THE SMITHSONIAN INSTITUTION, 

 APRIL, 1871. 



