592 SCIENCE PROGRESS 



much more space than was originally allotted, I am obliged 

 entirely to abandon many important propositions — concerned 

 with rational, equal and unreal roots, Lagrange's, Horner's, 

 Weddle's, Graffe's, and other methods, Tschirnhausen's process, 

 the algebraic study of the inverts themselves, much work on 

 partial roots, and above all the relation of the inverts (<£ _1 and 

 i|r _1 ) to the general operative multinomial theorem (<f> n and yfr n ) 

 — not to mention the solution of functional, difference, differ- 

 ential, and integral equations by operative division (see Verb 

 Functions, p. 71). A few notes are, however, added at 

 the end. 



The great length of the article is due to the necessity of 

 touching not only upon arithmetical iteration and critical 

 points, but also upon the o-algorithm. But regarding this last 

 matter, the article does no more than put it to the most 

 elementary uses, and no further attempt is made to indicate 

 the general employment of the algorithm either in Algebra 

 or Geometry or in the Calculus. 



The subject of this article is of interest, not only in con- 

 nection with the Theory of Equations, but because the whole 

 of Algebra from multiplication and involution upwards is 

 based upon operative involution (iteration) and because our 

 results emphasise the fact that the processes of formal or pure 

 operative algebra divorced from subject are simply mechanical 

 — in inversion as when direct — see Part II, p. 403. 



Our principal result has been, I think, that iteration and 

 operation division yield somewhat new methods for generating, 

 transforming, and summing certain infinite series and for 

 determining their ultimate convergence. How many series 

 belong to this class remains to be seen. 



I am indebted to Mr. Walter Stott and Mr. P. E. B. Jour- 

 dain for references to certain literature on arithmetical itera- 

 tion (Part I) ; but this cannot be examined in time for this 

 paper, and I hope therefore to deal with it in a subsequent 

 note. My article in Nature, October 29, 1908, was written 

 quite independently, and the method of solution there described 

 has been ably set forth in Dr. W. P. Workman's Memoranda 

 Mathematica (Oxford : Clarendon Press, 191 2, p. 46). 



