OPERATIVE DIVISION 577 



whatever value within limits x may have). That is, X is the 

 sum of x and all the iteration-ordinates. Now draw chords 

 through the summits of successive pairs of these ordinates and 

 let / be the tangent of one of these chords. Then 



K =-= (y n +i — yn)/(x n+ i - *„) = yn+i/y n - i = A*x n /Ax n . 



These chords constitute a polygon which cannot be crossed 

 by the curve / between x and X unless fx is somewhere less 

 than t ; that is, X always lies outside the polygon if fx > t. 

 And if y n + J and y n have the same sign, t n > — 1. The study of 

 the case when they have different signs (alternating iteration) 

 must be left to the reader — the condition for convergence is 

 briefly that fx > — 2 between x and x, when these are on 

 different sides of X — see (1) above. 



The root is therefore always attainable by iteration, either 

 by taking the arbitrary iteration-tangents less than the root- 

 tangent, or by using a vicarious operation with a root-tangent 

 greater than the proposed iteration-tangents. Thus in o ± // m, 

 the use of m may be interpreted in two ways ; either }/m is a 

 vicarious operation with a suitable root-tangent of about — 1 ; 

 or m is a suitable iteration-tangent. In the former case, in 

 fact, we use a vicarious curve with a flatter trajectory at the 

 root. Thus in Part I, Section III (2), p. 228, we saw that 

 0=1 — x — x 2 is not suitable for simple ascending division 

 (because the root-tangent is — 2*236) ; but on p. 230 we 

 rendered it so by substitutions which gave a flatter curve with a 

 root-tangent of — 0*28. The same simple explanation holds 

 for the results of Part II, pp. 406, 407. 



(4) Newton's form (Section V) may be called simple tan- 

 gential iteration {figure 5). In it we change m at each step 

 by putting m— fx n . It may go wrong at first because j/f 

 becomes infinite at least once between each pair of roots of 

 / ; but it is quickest at the root if this is single because then 

 f/f = 1 and FX = o. If the roots are multiple, I suggest 

 7=o —rf/f where r is the number of equal roots in the 

 group being investigated. For suppose that /= yjr r .x> where 

 •ty? consists of r equal factors of the form a — and % is the 

 product of the remaining factors. Then 



/' = ^-* . (r X + f . x') ; 



f'= V' 2 • {r(r - i)x+2rf. X '+W-x"}- 



