316 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



by 3n\ the denominators by 3n, and hence, for the value of — — ■ — there 

 comes out the progression 



+ 1 _0_ 3n 3 +2 81n 9 + 135?^ + 63^+7 

 -1' 0' 3?i 3 +l' 8l?i 9 +108?i 6 + 36?i 3 + 2' 



2 187^ 15 + 5 832w l2 + 5 589?i 9 +2 295?t 6 + 369^ + 15 

 2 187w 15 +5 103« i2 +4 131» 9 +1350?i 8 + 153ti + 3 ' 



the multipliers for which are 



r = l, q=-3, p = 27n 6 + 27n 3 + 3, 

 V(n 3 — 1) 



and similarly we find for ±L ^~ , the multipliers to be 



r=l, q=-3, p=272 6 -27n 3 + 3, 

 with the initial terms 



-1 _0_ 3n 3 -2 

 + 1 ' 3n 3 -l : 



these results agreeing with what has been found in the cases of 9 and of 7. 



It may also be observed that each third term of these second series is 

 reducible by 3, and that they form a progression converging still more rapidly. 



When the proposed number differs from a complete cube by more than 

 unit the extraction of the root is more complicated. As an example, we shall 

 take the number 3. 



In the equation « 3 — 3£ 3 = 0, on putting a—lb + c we get 



or 



-2b 3 +3b 2 c + 3bc' i + lc 3 = 0, 



fi2_!_&2 c _!_j c 2_-|. c 3 == (), 



whence the multipliers 



or, more conveniently 

 from which the progression 



r- 2"-. 9= T , PS 



r=-8>9 = l[>P=-2> 



— ^ ®l 303_ 1 371 6 199 28 035 



' ' ' 2 ' 4 ' 8 ' 16 ' 32 ' 64 ' 128 ' ' 



