58 AN ESSAY ON THE ROOTS OF INTEGERS, 



(14.) For the same reasons as in par. 10, 1 shall suppose the present 

 operation to be performed on a number whose 6th Root consists of 6 

 figmres. Let then M be a surd to the 6th power, and let its approximate 

 6th Root be m, so that m 6 Z_ and (m -J- I) 6 > M. Then since m consists by 

 supposition of 6 figures, so M will contain not more than 36, nor less 

 than 31 figures. 



If not, then either M contains fewer figures than 31, or more than 36. 



First, let M contain fewer than 31. Now since m contains 6 figures, so 

 by Lem. 5, m 6 contains at least 6x5 + 1, or 31 figures, which is absurd. 



Second, let M contain more than 36. Now since maximum of m by 

 Lem. 4 is W 6 — 1 so maximum of m -j- 1 is 10 6 and hence maximum of 



6 



(m + I) 6 is 10 6 I or 10 3<s which by Lem. 2 is the least number that can 

 contain 37 figures. But MZ-(m + l) 6 by supposition. That is, M must 

 always be less than the least number with 37 figures, and, consequently, 

 cannot contain more than 36. 



(15.) As a medium, let us suppose that M contains 33 figures, then by 

 the known properties of the series of par. 1, n will there be = 32, and M 

 may be thus represented, supposing the coefficients of the powers of 10 to 



be Digits. 



a- 10 3S + t>> 10 3! + c 10 30 + d- 10 a9 + er 10* 8 +/ 10* 7 +g- 10 s * 

 + Iv 10 ss + v 10 a4 + f 10 s3 + k' 10" + I- 10 ai + nv 10 i0 + n- 10' 9 

 4. p - 10* 8 -j- q- IO 17 + r- 10 IS 4- s' 10 ,s 4. f 10' 4 4- w 10' 3 4- v 10 ,a 

 + w 10" 4- x' 10 iq + y 10 9 4. z- 10 8 4- a- 10 7 + 0- 10 6 + y. 10 s 

 4. S' 10 4 4. g. 10 3 4- £• 10* + n. 10 4- 6. 



