THE SOLUTION OF EQUATIONS BY 

 OPERATIVE DIVISION.— Part III 1 



By SIR RONALD ROSS, K.C.B., F.R.S., D.Sc. 



CONTENTS OF PART III 



VI. (i) Another Proof of the Iteration Theorem — (2) Approach to 

 a Common-point of Two Curves — (3) Axial Iteration — 

 (4) Newton's Iterand — (5) Vicarious Operations, their 

 Root-Tangents, Mid- Axial Iteration, and other Forms. 

 VII. (1) Partial Roots— (2) Critical Points— (3) Examples— (4) Order 

 of Critical Points and Roots. 



VIII. (1) The Various Inverts given by Operative Division compared 

 — (2) Each Invert is an Operation performed upon a 

 Critical Point and refers to the same Root— (3) Each 

 Invert is also generated by a Rational Iteration based 

 upon the same critical point— (4) also by other iterands 

 — (5) The same for Ascending Inverts— (6) Ultimate Con- 

 vergence— (7) Proximate Convergence— (8) Conclusion. 



Notes.— I. The ^"-Fallacy— 1 1. Relations between Successive Roots 

 and Critical Points— III. Sterile Tracts— IV. Analysis by 

 Partial Roots — V. Critical Coefficients — VI. Remarks on 

 the Tables. 



Examples. — O TO R. 



Tables— I TO VI. 



Figures. — 1 TO 9. 



VI. (1) I owe my profound apologies to the readers of 

 Science Progress for the length of this article — due to the 

 necessity of writing it d plusieurs reprises. It is better, how- 

 ever, to trespass still further on their indulgence than to leave 

 it incomplete. 



The use of Taylor's theorem for the proof of the law of 

 convergence of iteration given in Part II, p. 410, may be 

 avoided as follows. We have, where D = d/ do, 



D [0 + f]x = [1 + f']x = 1 + f'x ; 



Z)[o + /]% = [i+/'> .[i+/'][o + />o = (itAo)(i4-N; 



D[p + f] n x Q = ( 1 + fx ) ( 1 + /'*,) ( 1 + /'**) . . . ( 1 + /'*„_,). 



All these factors will be numerically < 1 if all the numbers 

 f'xo, fxi, . . . are < o and > — 2 ; and if this is the case 

 D [0 + f] n x will become infinitely small as n is indefinitely 



1 Parts I and II appeared in the two previous numbers of Science Progress. 



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