CUBIC EQUATIONS BY HELP OF RECURRING CHAIN-ERACTIONS. 317 



the numerators being got from the multipliers 4, 6, 3, while the denominators are 

 powers of 2. From this, since a = b + c, we have the approximations to JS 



(0) (1) (2) (3) (4) (5) (6) (7) (8) (9) 



1 j5 21 97 437 1977 8 941 40 433 182 853 

 "0 ' 1 ' 3 ' 15 ' 67 ' 303 ' 1 371 ' 6 199 ' 28 035 ' 126 783 ' 



(10) (11) (12) 



826 921 3 739 613 16 911777 

 573 355 2 592 903 ' 11 725 971 



, &c. 



Here, as in the preceding cases, each third term may be simplified,. the pro- 

 gression being 



7_ 659 60 951 5 637 259 

 1 ' 5 ' 457 ' 42 261 ' 3 908 657 ' ' 



for which the multipliers are r = 64, q— — 48, /> = 98. 



Here the approximation is comparatively slow, the less accurate terms being 

 largely combined with the more accurate ones. 



To carry Brouncker's process one step farther, let us try how often b con- 

 tains c; for b = lc , the above equation gives + 5c 3 instead of zero; for b = 2c, 

 the result is + 3c 3 ; but for b = oc, we get - 17c 3 , wherefore b contains c twice 

 with something over ; we therefore write b = 2c + d. The substitution gives 



+ 3c 3 - 9c 2 d - 9cd 2 -2d s = 0, 

 or 



c s -3c 2 d-3cd 2 -^d s = Q. 



The multipliers hence resulting are 



2 

 r = -g, 2 = 3, p = 3, 



giving the progression for d : c 



, , 1 , 3 , 12 , 45-| , 175 , 670 , 2 565-i , &c, 



but a — dc + d, b = 2c + d, wherefore the ratio of a : b is given by the pro- 

 gression 



