OPERATIVE DIVISION 579 



Root-tangents are always required to test convergence and 

 can generally be put in simple forms owing to the disappear- 

 ance of part of the tangential at the root. For example 



£>[o+o//o :f]X = 1 +f'X.Do//o:o; 

 D [olio :QX=f'X. Dolfo : KX (or |X). 



The iteration is quickest near the root when I'X= o, and 

 we can select the parameters of an iterand so as to give 

 this result. For example let I = (£ -f- *»)/(£ + m). Then 

 I'X = i + Xf'X/tfX+tn)) and if this = o,m = -gX+XfX). 

 In using this fractional iterand, we obtain the odd positive 

 roots by putting the absolute term (positive) in £, and 

 the even roots by reversing the fraction. We can also put all 

 the terms with one sign in £ and those with the other sign in £. 

 Legendre's fonctions omales are of this class, but he does not 

 suggest the modifier m, which is necessary for rapidity. 



Many vicarious operations may be interpreted in terms 

 of iteration-tangents. Thus I = o£/(? may be written 

 / = + o(f— f)/£ ; showing that the iteration-tangent from 

 £ is zero, while that from £ is £/o. 



Other settings which may be occasionally useful for trans- 

 cendental and high-power equations, especially with the aid of 

 partial roots, are 



I=oVi± //>, rX=i± mXf'X/fiX ; 

 /=o+ mlog£/£, I'X= 1 + mfX/pC (or %X). 



We may add many exponential, trigonometrical forms, etc. 



VII. (1) When an equation fx = o is arranged in the form 

 £x — %x = o, each arrangement of this kind may be called a 

 setting ; £ and f may be called partial operations ; and the 

 roots of %x = o and of %x = o may be called partial roots, 

 the roots of }x = o being the actual roots. Now definite rela- 

 tions exist between the partial and the actual roots — relations 

 which often serve to locate the latter easily and closely. I do 

 not know whether the subject has ever been fully treated, 

 but it is suggested by the fact that the inverts given by opera- 

 tive division are always functions of partial roots ; and it must 

 therefore be briefly dealt with here. 



A partial root may be said to refer to an actual root when 

 no other actual root or other partial root of the same setting 

 comes between them. Then the following propositions can 



