308 Mr. J. J. Sylvester : Meditation on the 



we may easily establish the following theorem, viz. that 

 r 8 + N r* + 3rN r^ + fo^N + N 2 r 5 +10r 8 N + 5rN 9 '* 

 2r ' 3r 2 +N' 4^ + 4r^i ' 5r 4 +10r 2 N~+N 2 ■ ' " 



will be successive approximations to VN, whose limits of error 

 can be assigned when a limit to the error of the first approxi- 

 mation r is given. The coefficients of the qth approximation, it 

 will be observed, are for the numerator the alternate binomial 

 coefficients 



1 c £zi Q (£=1>. fe=?). «=? & 



i, q 2 > g- 2 3 4 ,«c, 



and for the denominator the intermediate ones, 



q-1 q-2 9 



Mr. Cayley has reminded me that the third approximation, 



— -g — ~-, is a special case of a formula for any root of N given 



in the books ; and to Mr. De Morgan I am indebted for a hint 

 which has led me to notice that all these forms may be deduced 

 from the Newtonian method of approximation f. 



If we call the ith approximation <f>(i, r), we shall find that the 

 functional equation <f>(j, <f>(i, r]) = <f>(ij,r) will be satisfied; 

 which is not so mere a truism as might at first sight be supposed, 

 as any one may satisfy himself by studying the analogous theory 

 for cubic or higher roots, a part of the subject to which 1 may 

 hereafter return. 



Now as to the limits of accuracy afforded by the successive 

 approximations. Let e be a known limit to the relative error of 



the first approximation r, by which I mean that ( =J- J <e 2 . 



• 



* In other words, if r be the first approximation to VN, the »th approxi- 

 mation will be 



( r+ VN)*+(r-VN) f V5r 



(r+VN)'-(r-VN)« ' 



so that the relative error becomes 



2(r— VN)' 



(H- V1T)«— (r-VNy' 



in which form the theorem is self-subsistent, and needs no proof. But the 

 fact remains interesting, that the application of Newton's method of ap- 

 proximation to the equation # 3 =N will be found to lead to the form above 

 written at the »th step of the process conducted after the continued-fraction 

 fashion. 



t The expansion (after Newton) of VN introduces the binomial coef- 

 ficients — a curious fact ! What are the analogous integers which the con- 

 tinued-fraction process applied to \/N will produce? 



