OPERATIVE DIVISION 405 



It suffices to record integers only until we approach quite near 

 the root. In this case, starting at 100, Newton's method is 

 slower than Dary's, but outstrips it near the root. In both 

 methods we shall save labour by commencing as near the root 

 or roots as possible. In Newton's method, if we commence 

 with xq = 0, or = — 100, we shall obtain quite divergent series. 



One reason why both methods receive such scanty ex- 

 position in the books is that while both may succeed at times 

 both may also, perhaps more frequently, go wrong ; and it 

 cannot be said that subsequent writers (whose works I have 

 seen) have altogether shown how to apply the methods, first 

 with certainty, and secondly with the quickest possible results — 

 much less how to obtain algebraic expressions for the series 

 which the methods produce. It is time then that these questions 

 be considered. 



(2) The process of successive substitution employed in 

 these methods is obviously the same as that of operative volution 

 or iteration defined in Section I (4). For if 



Xi = [<f>]x x-2 = [<f>]xi ... x n = [<£]#„_! ; 

 then x n = [<f>] n x . 



In such cases we may speak of the iteration of <f> upon the base 

 Xq ; and may employ the names iterand for <f> and iterants for 

 the series of operations <£, $% </> 3 • • •, or for the series of 

 numbers x , X\, x 2 . • • 



The methods of Dary and Newton are special cases of a 

 more general and complete theorem which I have not yet seen 

 fully stated, but which may be put as follows. 



First, if the equation o = fx possesses only one real root 

 x = X between any two values of x, x = a and x = J3 ; and 

 if / is continuous between a and /3, and fa is positive, and 

 x is any real quantity between a and /3 : then the iteration of 

 + / upon # will infallibly approach the root X if the tangents 

 fx are always greater than — 2 for values of x between a and 

 /?. And if the tangents fx are always greater than — 1 for 

 these values the iterants xo, X\, #2, . . . will successively increase 

 if x <X and will successively diminish if x >X, until in both 

 cases the root is nearly reached ; and if the tangents f'X lie 

 between — 1 and — 2 the iterants will ultimately be alternately 

 less than and greater than the root, but will still approach it ; 

 and the iteration will be quickest near the root if f'X = — 1 . 



