57 8 SCIENCE PROGRESS 



When these operate on X the tyX factors vanish, and since 



l'X= i-r[i-f/f .f"/nX, 

 this becomes i — r{ i — (r — i)/r} = o at the root. For 

 example, for the two equal roots of x 3 — 3* + 2 = o, 



7=0-2///= (o 3 + 30 - 4)/(30 2 - 3), 

 and we have the iterants o, 4/3, 64/63, 24004/24003 . . . ; 

 which scarcely support the common statement that Newton's 

 method is useless for multiple and for commensurable roots. 



Fourier's rule for applying Newton's method merely amounts 

 to this, that if x be taken on that side of X at which / is convex 

 to the axis of x then the iteration will be progressive. But 

 this is by no means always obligatory ; for if x Q be taken on 

 the other side of X but sufficiently near to it, then x y will 

 generally conform to Fourier's rule. Newton's is the most 

 valuable method of arithmetic iteration we possess for equations 

 of most kinds provided that x is near to X. Both it and the 

 previous method can generally be worked throughout with 

 rational figures. 



(5) Arithmetical iteration is not the subject of this paper ; 

 but in order to survey the subject in gross I will add a few 

 remarks about other vicarious forms as defined in Section V (2), 

 p. 406. These forms are generated as follows. If fX = o, 

 then 0//0 :fX=o. If £X - £X = o, then 0//0 : (£X -£X)=o; 

 0//0 : %X — 0//0 : %X= o ; and indeed <f>£X — <f>£X = o, where 

 0//0 and <j> can have many values. The parameters of the 

 operations 0//0 or <f> may be constants or functions of x, so 

 long as we do not thus introduce new roots which clash with 

 the original ones. These theorems contain of course the funda- 

 mental properties of equivalence. 



If in the equations of subsection (2) above, £= 0, m = o, and 

 n = 1 , then we have the case of mid-axial iteration in which 

 7= + f — 0= £ (figure 6). This is a form much discussed 

 by previous writers, including myself ; and Dary's form given 

 in Part II, p. 404 is an example. We put the original equation 

 in the setting x= £x — which can be done in many ways — 

 and then find the common-point of the mid-axis (or y = x) 

 and £. The tangential form of it, namely 1= o + (£— 0)/ 

 (1 — £') is quick near the root since ]'X= o. Mid-axial itera- 

 tion has advantages in some special cases and disadvantages 

 in others. 



