ESSAYS 



OPERATIVE ALGEBRA: OPERATIVE INVOLUTION (Colone 

 Sir Ronald Ross, K.C.B., K.C.M.G., F.R.S.) 



10. In 1905 I described a method for the explicit rendering of algebraic opera 

 tions apart from their subject (1). So far as I know, no attempt of this kind 

 had been made previously, and I therefore called such expressions Verb- 

 Functions ; showed that they produce & formal algebra of operation apart from 

 quantity ; developed some of its elementary rules, homologous to those of 

 numerical algebra ; and illustrated the subject by using " operative division " for 

 the solution of algebraic and differential equations (1, 4, 5, 6). 



In 1908 I described briefly another method for resolving numerical algebraic 

 equations, namely by iteration — a method which was subsequently found to have 

 been originated by Michael Dary, a friend of Newton, in or before 1674 (2, 3). In 

 191 5-16 I resumed these subjects in Science PROGRESS (4, 5, 6), and showed that 

 the solution of equations by operative division provides the algebraic expression ot 

 their solution by arithmetical iteration. I now give some fundamental theorems 

 of Verb-Functions, mostly connected with "operative involution," and all taken 

 from a lengthy paper completed in July 1916, but not published owing to war-work 

 — of which paper this one is an abstract.* 



It should be added that the notation employed really presents no difficulty, 

 while it greatly abbreviates and facilitates the whole algebra of substitution by 

 bestowing upon it a formal process like that of algebra. And practically no other 

 notation can be employed if we wish to denote operation apart from subject (1, 4). 

 We merely define that O n is an operator which raises its subject to the nth 

 algebraic power, and that an expression within square brackets is an operator 

 which operates on the matter which follows — as for example in 



[O n ]x =x n ; [a + bO m - cO r ]x = a + bx m - cx r . 



An index outside square brackets denotes that the operator within those brackets 

 is to undergo operative involution (also called iteration) or evolution, as the case 

 may be. But this notation will be familiar to those who have read the previous 

 papers in Science Progress (4, 5, 6). It is really based on recognition of the 

 fact that 4>*, /\°, 2°, -D , etc., are not equal to numerical unity, as generally sup- 

 posed, but to the operative unit or symbol oj substitution, O. The error that <f>° = 1, 

 A 0=I 5 and so on, has largely vitiated symbolism in the past ; and the correction 

 of the error now renders this symbolism as rigid as that of ordinary algebra. Note 

 that O is not, of course, a number but an operation ; that O = 1 ; and that, when a 

 is a number, [a]x = [aO']x — ax* = a. The square brackets may be dropped for 

 purely operative symbols, as in writing cj>-^r and 4> n f° r i^W an d [<£]" ; but then 



* Much work was done by me years ago on the application of Verb-Functions 

 to the Calculus, by studying the iteration of operations which vary from step to 

 step —by which I deduced a Calculus of which the additive Calculus generally 

 used is only one case. I also applied them to Determinants, Eliminants, etc., but 

 these researches cannot be published till this one is. 



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