57 6 SCIENCE PROGRESS 



straight lines — though if they are not, J'X will have a some- 

 what different form. Observe also that the iterand may be 

 varied at each step, even for the same root ; for in the 

 expression 



■wr %r %r-\ %1 X\ 



-A = = = . . . = = #o 



each operative factor may be different. The first theorem is 

 included in the second. 



The equation jX = o can be set in the form £X — %X = o 

 in many ways. If / = £— £, then the iteration will be pro- 

 gressive if j'x >m — n between x and X — since f = £' — £'. 



(3) The simplest setting gives what may be called simple 

 or axial iteration. In this we take £ = / and f= 00 (that 

 is, the axis of x itself), while n =0. If m = — 1, we have 

 the unmodified iteration of o + /, in which each iterant Xi is 

 found by drawing a straight line from the summit of the 

 ordinate jx downward at an angle of — 45 ° till it meets the 

 axis of x (figure 3). If m is some other negative number we 

 have the modified iteration of o — // m, in which the straight 

 line is drawn downward from the summit of the ordinate fx Q 

 at some other and more convenient angle. Thus m should be 

 numerically large if X is near to x , and, ceteris paribus, smaller 

 if it is further (figure 4). If m is numerically large, the iteration 

 will be more slow, but more sure ; and if it is small the 

 iteration may become alternating. In both cases, for the 

 most rapid approach near the root we should there have 

 m = f'X as nearly as possible. If / is continuous and /(o) is 

 positive, m must be negative for the odd positive roots and 

 positive for the even ones — as j'x is. See also Part II, 

 p. 406. 



The condition for progressive convergence is that j'x be 

 never algebraically < m between x and X — though it may 

 be as much greater as we please ; but the reader will do well 

 to examine this particular form of iteration for itself. Let 

 y* - J x n De called an iteration-ordinate (unmodified) — so that 

 *n + i = x n + y n . Then 



X = * + yo + y x + y* . • • y x ; 



(and we note in passing the bearing of this theorem on the 

 general subject of convergence, since the same result is reached 



