320 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



Let us take the general case of a number exceeding a perfect cube by, say, 

 a ; that is a number of the form n 3 + a . We have here a = nb + c, and the 

 equation 



a s -(n s + a)P=0 



becomes 



- ab 3 + 3n 2 bc + 3nbc 2 +c 3 =0 , 



which gives the multipliers 

 or more conveniently 



- 1 ■ = 3 ' 1 - =^!i 2 



a a a 



a 2 3an 3n 2 



a 8 or a 



from which we have the elementary progression 



x 3n? 9?i 4 + 3cm 27n 6 +18g7i 3 + a 2 &Q 

 a a 2 a 3 



and thence the progression of fractions 



cT_' 0_ 3ft, 3 + a 9n s + G a n 2 27 "n 6 + 27 an 3 + 4q 2 



.0 ' 3n 2 ' 9n* + 3 a n ' 27n 6 + 18an 3 + a? ' 



following exactly the law of those already found when the excess a is unit ; the 

 multipliers being a 2 , 3an, 3?i 2 . 



By a proceeding exactly analogous to that formerly used, we find that the 

 convergence to the cube root of the ratio n 3 — a : n 3 , is obtained from a progres- 

 sion of which the multipliers are 



r = a 6 ; q=-3a i ; p = 27n e + 27 an* + 3a 2 , 



the initials being; 



+ q- 0. 3n 3 + 2a 

 -a' 3 ' 0' 3n 3 + la' 



This formula may be generalised by substituting for n* any number K. Then 

 lj 1 T a is obtained with the multipliers 



« 6 ; -3« 4 ; 27K 2 + 27aK + 3a 2 



from the initial terms 



-fa' 3 . 0. 3K + 2« 

 -a" 3 ' 0' 3K + lo' 



