Idea o/Toncelet's Theorem, 309 



For greater simplicity, I take separately the cases where r is too 

 great and r is too small. 



1. Let \/N<.r< (1 + € ) ^N ; then the errors will be through- 

 out in excess ; and we may assign as a limit of error to the ith 

 approximation a quantity, say e i} which is a known function of 



2 . . 6* 



e, viz. /n~~i — fv — v wn ^ cn *t mav ^ e not i ce d is less than ^r~^, 



2. Let \/N>r>(l— rj) -i/N; then the errors will be alter- 

 nately in defect and excess, and to the zth approximation we may 



assign a limit of error rj if where 9? f = g -i _iv _/ -i\i*- 

 We may now apply these results to Poncelet's linear approxi- 



* If we write 



€ { — ${€,{) and ?; £ =%,«), 



then if « be any odd number, 



6(d( € ,i)J)=0(e,ij), 

 S($(ri,i),j)=S(ri,ij); 

 and if i be any even number, 



*(*(«, *).i)=*(*V)> 



Or more simply, if the error in excess be treated as positive, and in defect 

 as negative, and 8 be the first and 8 { the ith limit of error, we shall have 



and calling 8; —6{i, 8), 



N+a? 2 

 Thus, then, if we call — g — =\jrcc, yjr q x will correspond to the (2 q )th 



order of approximation, and the absolute value of the error w,ill be less than 



2»* 



(2+bf-lF 



By way of example, suppose we take 6 as our first approximation to v31, 

 then 



and if we make ylrx=z J~ x , we shall have 

 2a? 



* Vr 4 6: V31::l+«:1, 



where 



2 



G>< 



23 i6 — r 



which serves to exemplify the prodigious rapidity of the approximation in 

 this method of extracting the square roots of numbers. 



