59 o SCIENCE PROGRESS 



(6) Lastly we have to enquire when each invert is conver- 

 gent. We have seen that all the inverts are generated by 

 various iterations — the corresponding geometric construc- 

 tions of which are obvious and easy (Section VI) ; and may 

 therefore suppose that the conditions of convergence will be 

 equally clear. But it will be asked at once how this can be 

 the case, since different iterations, even towards the same 

 root, have different conditions of convergence. The answer 

 appears to be briefly that most of the generating iterands are 

 irrational, as they consist of roots or ratios developed in infinite 

 series which themselves have conditions of convergence in 

 addition to those demanded by the iteration. I have space 

 therefore to deal only with the rational iterands, namely 

 o - f n /qg n ~ l and o + (f> /b. 



An iteration may diverge either at the critical point from 

 which it starts or at the root which it should reach — in both 

 cases owing to the use of an unsuitable angle of iteration. We 

 may therefore talk of ultimate convergence (at the root), and 

 of proximate convergence (near the critical point or base). 



Regarding ultimate convergence, the iterand gives immediate 

 and definite information, concerning both arithmetical and alge- 

 braic iteration — information which cannot easily be otherwise 

 obtained. For since the series represents the arithmetical 

 iteration we may, I suppose, assume that the former will be 

 ultimately convergent or divergent when the latter is so. Now 

 by Sections V and VI, the iteration of / = o — fjqg n ~ l will 

 succeed if I'X lies between -f- i and — i , and will be most 

 rapid near the root if l'X=o. That is, i — f' n /qg n ~ l must 

 lie within these limits ; that is, qg n ~ l must be greater than 



-f' n X and should nearly = f'nX. 



If we do not use a modifier, then g = K q , the critical point, 

 and we may not be able to fit a convergent iteration to it. 

 But by the use of the arbitrary modifier m we can adopt 

 such an angle of iteration as will ensure both the arithmetical 

 and the algebraic iteration converging at the root, and shall in 

 fact employ the simple modified iteration described in Section 

 VI (3), which can be always made to succeed. We find m 

 from the equations (p = 1) 



g= Vy= Vm + K* = n -Vf' n X/q, 



