OPERATIVE DIVISION 591 



for the quickest possible iteration at the root. Of course we 

 do not know f'X exactly ; but we can always roughly estimate 

 its magnitude sufficiently to ensure that g shall be large 

 enough. 



For the simple ascending iterand o -f- 4>o/b we observe that 

 b = </>' (o) — the tangential of </> at the point zero. This 

 should be nearly the same as the tangential at the least 

 positive root <f>' X, and will be so if we move the origin suffi- 

 ciently close to that root — which is the simple explanation of 

 the rules given in Part I, Section III. 



Applications are given in the Examples. 



(7) Hence we may apparently be certain that if the ulti- 

 mate convergence of the arithmetical iteration is assured the 

 corresponding operative invert will really and correctly repre- 

 sent the required root if the whole of the series be considered. 

 But this does not mean that the first few terms of the series 

 need necessarily approximate, even roughly, to the value of 

 the root ; not even indeed when the generating arithmetical 

 iteration does so. For in the series the terms of Ig, Pg, Pg. . . . 

 are rearranged under progressive powers of g. All the terms 

 of Ig, namely, 



ig = g - Pi/q - Pz/q • g~ l - Pz/q • g~ 2 - • • • 



(for the equation x n + piX 11 ' 1 + . . . + p n = 0) are generally, 

 it is true, contained in the first few terms of the invert ; but 

 if we take only these first few terms, the higher terms of Pg, 

 Pg, . . . will be cut off, and the beginning of the series may not 

 be sufficient. 



It comes to this, then, that, when we wish to obtain an 

 approximation from the first few terms only, then the value of 

 g in comparison with the coefficients p lf p 2 , p 3 , . . . should be 

 as large as possible for descending inverts and as small as 

 possible for the ascending ones — that is, the successive major 

 terms should diminish rapidly. Or else the coefficients of g 

 in the invert should do so. These matters will be further dis- 

 cussed in Note VI. 



(8) Conclusion. — I have now concluded my task of endeavour- 

 ing to show how each invert generated by operative division 

 is also generated by iteration upon the proper critical point — 

 thus indicating when each series is ultimately convergent and 

 to which root it approximates. But, though I have occupied 



