318 



EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



(0) (1) (2) (3) (4) 



(5) 



(6) 



(7) 



0^ 

 ' 



o 



2 ' 



10 



7 ' 



39 



27' 



149 



570| 2 185 



8366J 

 1515' 5 8001' 



(8) (9) (10) (11) 

 32 034$ 122 659 469 657| 1 798 306f f 



(12) 

 6 885 667f 



22 211* ' 85 047 ' 325 622if 1 246 806^1 ' 4 774 255 



, &c, 



converging much more rapidly. 



Lagrange's application of Brouncker's method may be still farther con- 

 tinued, as in the following scheme : — 



0= a 3 - 0a 2 b- Oab 2 - 3 ¥ 



0=- 2b 3 + 3b 2 c + 3bc 2 + lc 3 

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

 0=-29d 3 + 18d 2 e + 18de 2 + 3e 3 

 0= + 10e 3 -33c 2 /-69c/ 2 -29/ 3 



a = lb +c 

 b = 2c+d 

 c = 3d + e 

 d~U+f 

 e=4:f+g 

 &c. 



until we arrive at some equation promising greater facility or greater rapidity. 

 The last of the above gives 



r = 2.9 ; ? = 6.9 ; p = 3.3 ; 



with the condition a = 13e + 10/; b = 9e + 7/ 



For the cube root of the next number, 4, we have 



a 3 - 0a 2 b- 0a& 2 -46 3 =0, 



- 3b 3 + 3b 2 c + 3bc 2 + lc 3 =0 

 + 4 c 3 - 0c 2 d- 6cd 2 - m 3d 3 =0 



- 5d 3 + 6d 2 e+ 12de 2 + 4e 3 =0 

 + 12 e 3 - 24e 2 /- 24c/ 2 - 5f 3 = 



- 53/ 3 + 24/ 2 ^/+ 48/^ + 12^ = 

 + 31g 3 - 63g 2 h-135gh 2 -53h 3 =0 

 -188h 3 + 324h 2 i + 2Wd 2 +31i 3 = 



&c. 



The equation in e and /put in the form 



Cl = lb + C 



b = lc +d 

 c = ld + e 

 d = 2e+f 

 e=2f+g 



g=3h + i 



h=2i + Jc 



&c. 



e 2 -2e 2 f-2c/ 2 -^f 3 =0, 



5 

 gives the multiplications p=q7 » <7 = 2 , P = 2 > an d these, along with the con- 

 ditions 



a = 3e+3/; Z> = 5e+2/; 



