1826.] theUseofcojUirmediyactions,&^c, 4^ 



-,=^ <,.mar (9) 



^ C nP+ n + m.x ^ 



8 + ^r- 2n-m.x ^ 



3„p + 2n+wi.a; 



^ "*" 5nP + &c. 



in »« 



The other variety is N^ = (P - x) " = 



■^ ^ r n U; + n.2n VN/ n.2n.3n Ui ^ S 



which reduces to 



IV.-.V (10) 



mx 



P" X 5l + — Tr »»-»»-^ ^ ^ 



C n N + n + w . X _ 



2 + -r rr- 2 n- 111 . X ^ 



3 n N + — 2 n + m . a: 



2 + ' , -— 



5 n N + &c. 



The first or second of these theorems may be used, according 

 as the assumed root is less or greater than the true. 



It deserves to be remarked, that Halley's rational method, or 

 the common rule for approximate evolution, is contained in the 

 first three terms of each of these formulae. The error, therefore, 

 of that method may be estimated by formula (7). 



Likewise the general form of the fraction equivalent to any 

 odd number of terms of the continued fraction will be found to 

 coincide with the formula for (« + by given by Euler in Inst. 

 Calc. Dift'. vol. 2, § 239, without investigation. 



The facihty of aggregating a continued fraction, and the. 

 opportunities it affords of simplifying its terms, and of making 

 allowance for the effect of the final portion which is omitted, are 

 peculiar recommendations of the praxis by the continued frac- 

 tions (9, 10) in preference to either of the other modes of 

 evolution. 



Example VL— Extract the seventh root of 2. By formula (9), 



wefind(l4-iy = 



, 1 6 8 13 15 20 22 27 



"*" f + 2 + 27+ "2~+ 35"+ "2"+ 49"+ 2~+ &c. 



~ ^ "^ 7 + 1 + 21 + 2 + 7 + 1 + 3/ 



27 



putting 1/ = the neglected part 49 + g- + &c» = 61 in the 

 nearest integers. Hence we have 



1 7 1 21 2 7 1 61 



1 1 8 11 263 /669_\ 223 f^^_\ ?^ *133 79267 



r 7 To 2*38 \606 "" y 202 V 16^ "~ / 826 l"028 71794 



1 3 4 13 (3) (2) 11 



Wherefore —^ = M0408947 is the rootrequired. 



Nev) Series, vol. xn. k 



