314 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



Hence for a progression converging to the ratio b : c we have the multi- 

 pliers r=lj ^ = 6, p = 12, giving the series 



, , 1 , 12 , 150 , 1 873 , 23 388 , &c. ; 



and hence, converging to the ratio of a : b , we have the progression 



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

 1 2 25 312 3 896 48 649 607 476 7 855 502 94 719 529 

 0' 0' 1' 12' 150' 1873' 23 388' 292 044' 3 646 729' 45 536 400' 



(9) (10) (11) (12) 



1182 754 836 14 768 960 708 184 418 777 041 2 302 821 843 576 

 568 609 218' 7100175 745' 88 659 300 648' 1107 081271464 



, &c. 



Here the terms (3) , (6) , (9) , (12) are reducible by the common divisor 6, 

 and form the progression 



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



_2 52 101 246 197125 806 383 803 640 596 

 -1' 0' 25' 48674' 94 768 203' 184 513 545 244' 



which proceeds according to the multipliers r = l , q— — 3 , p — 1947 . 



This convergence is so rapid that the error of the term (2) cannot be 

 detected by help of the ten-place logarithms ; that of (3) is beyond the 

 precision of the fifteen-place tables. 



In the case of the number 7 , which is less by unit than the cube of 2 , the 

 convergence is somewhat slower. For the equation 



a 3 -7P=0 



it is convenient to take the first measure in excess, and to write a = 2b— c, 

 which gives 



b 9 -12b 2 c + 6bc 2 -c 3 =0; 



so that the progression, by help of the multipliers r — \, q=—Q, p — \% 

 becomes 



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

 -1 2 23 264 3 032 34 823 399 948 4 593 470 

 ' 0' 1' 12' 138' 1585' L8 204' 209 076' 2 401273' 



(8) (9) (10) (11) (12) 



52 756 775 605 920 428 6 959 097 956 79 926 409 679 917 968 248 840 &c 

 27 579 024' 316 749 726' 3 637 923 841' 41782 166 760' 479 875 207 800' 



