of Irrational Functions by means of Rational Substitutions. 447 

 {n— l)a n +p 



na" 

 (n- 



(n — l){(n— l)a n +p} n + n n .p.a< n ~V 

 n 2 a n - 1 {(n—l)a n +p} n ~ l ~~! 



If we make ti=3, we have for the fractions converging to the 

 cube root, 



2a 3 +p 2{2a 3 +p) 8 + 27pa 6 „ 

 a > 3a* ' 9a*{2a?+p)* ' 



Applying this to the cube root of 2 by making a = 1,^=2, we 

 obtain successively, 



i > t^- 333 ' S= i ' 36389 > and iSSS-8= 1 - 3599335 ' 



the true value being 1*25992105. 



The mode of finding the limit of the error is given in the 

 books on the theory of equations. 



The above formula? obviously enable us to approximate to 

 2 

 functions of the form tyndx. In fact our first approximation 



would be I ; * dx, a and y being both rational func- 



J na n ~ l 



tions of x, the former chosen pro arbitrio. But we are by no 

 means restricted to pure radical forms. The process applies with 

 equal generality to the functions which are given implicitly as 

 the roots of equations. Thus, if the given equation be y m + \y= v, 

 a first approximation to \ y dx is 



J' 



ma m ~ l + X 



Here a and y are supposed to be rational functions of x, and \ 

 andj? to be constant. 



If we make n — 2 in the formulae for the simple radical, we 

 obtain the series of arithmetic means which I gave in the memoir 

 above quoted, viz. 



gt+p gt + GaZp+p 2 a s + 2Sa 6 p 4- 7(ky .+ 2 Sa, y +p 4 fr 

 a ' ~2a~~' 2a{a*+p) s "" 8a{tfi+7rfp + 7ay +f) > 

 The harmonic series of means is obtained by multiplying the 

 reciprocals of these into ap. 



"With reference to the question as to which method is the 

 most advantageous for the purposes of elliptic functions, it may 

 be observed that the labour attending the mth approximation in 

 my scale is about equivalent to that of the 2mth in Mr. Sylves- 

 ter's. It will therefore be a sufficient test to try his tenth and 



