Ross — Verb- Functions. 69 



(7) For further examples, it will be advisable to consider the 

 equations given in a text-book^ for illustrating the ordinary methods 

 of approximation. 



The equation x'^-2x-b = Q was I^ewton's example for his method 

 of approximation, and has a root = 2-09455148 . . . Here, p^i = - 2, 

 and = - 5. Evidently, 8 < ^/ 25, so that we must use the full form 



;r^-2^ = 5. As — — is little less than unity, the approximation will 

 be slow. Putting g for ^5 , we have, from the formula 



"^3^2 - 3 3 3 [3i^y 3 3 3 3 [4V/ 

 = 1-71 (1 + -22800 - -00395 + -00090 . . .) 

 = 2-09464. 



Only five places of decimals have been preserved, and a low approxi- 

 mation given to the value of ^5. 

 The equation 



2x^ - 473.^2 - 234^ - 711 = 



has been taken to illustrate Horner's method of approximation, and 

 has the commensurable root 237. The predominant coefficient is 

 evidently ^t-, and the rate of approximation high. Hence we find 

 at once from the form 



y - 468?/-i - 2844y - = 473, 

 where x = ^y, that 



^ = *[ 473 + lg + 1 = 1 (473 ^-989-^-012...) 



= 237-00 



The equation x^ - ^x"- - 2x + b = Q is used to illustrate Horner's 

 method for incommensurable roots. It has three which are given as 



3-128 . . . , 1-2016 , . . , and - 1-330058739 . . . 

 The coefficient 3 is evidently > v^S, and nearly as great as 



— O. 



4 



^ Todhunter's "Theory of Equations," 3rd Editiou. 



