574 SCIENCE PROGRESS 



increased. That is, the slope of the curve [o 4-/3°° will be 

 zero and the curve will be parallel to the axis of x, not only 

 for a specific value of x , but for all tracts of x within which 

 j'x < o and > — 2. The remainder of the demonstration is the 

 same as that briefly given in Part II, p. 410. 



If f'xo, f'xx, . . . are < o and > — 1, all the factors will be 

 positive and the iteration will be progressive. But if some 

 of these numbers lie between — 1 and — 2, some of the factors 

 will be negative, and the iteration may be alternating. In the 

 latter case, however, the alternate iterants, say x , x 2 , x iy . . . 

 will approach the root progressively from one side, while the 

 others, x u x 3) x 5 , . . . will approach it progressively from 

 the other side, and we have in fact to deal with the iteration, 

 not of o -+•/, but of [0 +/] 2 . This of course follows the same 

 rules ; but if the equation has more than one real root, x 

 must be taken so near to the required root that x x does not 

 surpass the other one. 



Observe that the earlier factors in the value of D [0 + fYx 

 do not count in comparison to those near the root ; for if the 

 latter are numerically < 1 the infinite iteration of them will 

 still ultimately reduce the slope of the curve to zero even if 

 many of the earlier factors are very large. Where the root 

 sought is nearly equal to the next one, the factors near the 

 root may be nearly unity, but for the proper root it must be 

 still < 1 . Where the root sought is one of a group of equal 

 roots, the factor at the root, namely 1 + f'X, will be unity ; 

 but the earlier factors will be < 1, and the equal roots will 

 lie near the mean of two iterations, one taken from above and 

 the other from below. Where 1 + f'X = — 00 (as in the case 

 of + /f where r > 1), or where it is < — 2, we can find vicarious 

 operations in which these difficulties do not exist. 



(2) Perhaps the most general proof of the law of conver- 

 gence of iteration is briefly as follows. Suppose that we seek 

 the abscissa X of the common-point of any two curves £ and £. 

 Let x be any abscissa near it. Then by Part I, Section II (7) 

 the operative ratios gx // x and £# //#o denote all the curves 

 which pass through the points (§k > *o) anc * (%x , x Q ) respec- 

 tively. Let two of these curves be straight lines with tangents 

 m and n to the axis of x. Then 



§*o // *o = &o — mxo -f rao %x // x = %x — nx + no ; 



