222 PROFESSOR THOMSON'S INVESTIGATION OF A NEW SERIES FOR 



By means of any of these six formulse, in connexion with series (11), the va- 

 lue of -^ may be computed with great despatch and facility. As an example, let 

 us take formula (18), and putting successively p = \, q = 7, and j9 = 3, q = 79, in 

 formula (11), we get 



_ r 2.8 + f Ax0.02 + f Bx0.02 + f CX0.02 + &C. ) 

 " ~ t + 0.30336 + f A' X 0.00144 + f B' X 0.00144 + &c. j 



Hence, by carrying the decimals out to twelve places, the computation will stand 

 thus : 



First Series. Second Series. 



A = 2.800000000000 A' = 0.303360000000 



B = 37333333333 B' = 291225600 



C = 597333333 C = 335492 



D= 10240000 D'= 414 



E= 182044 ~J 



p _ 33 ^ (J 8 tan-' ^= 0.303651561506 



G= 61 1 

 20 tan-*- =2.837941092081 



20 tan-* I = 2.837941092081 



' TT =z 3.141592653587 



This value of '^ is true in all its figures except the last. The computation of 

 the terms is effected with great ease. Thus in the first series, B is found by 

 doubling A, subtracting from the result one-third of itself, and rejecting the last 

 two figures ; C by doubling B, taking from the result one-fifth of itself, and re- 

 jecting two figures ; and so on : while, in the second series, B' is found by multi- 

 plying A' by 144 (which is easily done on account of the repetition of the figure 4), 

 by taking from the result one-third of itself, and rejecting five figures : and, in 

 both the computations, various arithmetical contractions will suggest themselves 

 as the work proceeds. 



Numberless other expressions for it might be obtained, and of any degree of 

 convergence whatever. Those given above, however, are preferable perhaps to 

 any others, on account of the simple form of k, and the consequent facility with 

 which it is managed in the computation. The following may be mentioned in 

 addition to those akeady given : 



3 OQ 

 ^ = 28 tan-'- -t- 20 tan''^^ (19) 



3- = 48 tan-* J- + 20 tan"* ^^ C20) 



79 22049 ^ ^ 



^ = 68 tan-*;^ + 20 tan-* -^- (21) 



79 873121 ^ ^ 



. = 22tan-*i + 2tan-*^^^^^ (22) 



/ 69049993 ^ ^ 



