THE SOLUTION OF EQUATIONS BY 

 OPERATIVE DIVISION. 1 — Part I 



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



CONTENTS OF PART I 



I. Prefatory. 



II. (i) Operative Notation— (2) Reversed Operation— (3) Operative 



Multiplication— (4) Iteration— (5) #° does not equal Unity— 

 (6) Operative Ratio— (7) Operative Geometry— (8) Operative 

 Division. 



III. (1) Inversion by Operative Division— (2) Trinomial Equations -(3) 



The General Rational Integral Equation by Simple Ascend- 

 ing Operative Division— (4) Expression of each Root by an 

 Algebraic Series— (5) Examples A to N— (6) Note. 



I. Thirty years ago I devised a notation which enables us 

 to express any algebraic operation without stating the actual 

 number upon which it operates ; and I called such an ex- 

 pression a Verb Function, because it denotes rather the action 

 taken in arranging quantities than the quantities themselves. 

 The notation adds greatly to our powers of mathematical 

 expression ; gives us a precise algebra of substitution ; and 

 enables us to render quite rigid the symbolic algebra which was 

 extensively developed by Boole for the Calculus. A lengthy 

 paper on the subject, called " Verb Functions with Notes on the 

 Solution of Equations by Operative Division," was published 

 ten years ago (Proc. Roy. Irish Academy, vol. xxv. sec. A, 

 No. 3, April 1905, pp. 31-76) — in which I described the 

 notation, the elements of the " operative algebra " which it 

 leads to, and also an application to the solution of algebraic 

 and differential equations which was, so far as I can ascertain, 

 a new method. Some years later I found independently 

 another method for approximating to the roots of numerical 

 equations by " iteration " (Nature, October 29, 1908) ; but 

 my friend Mr. Walter Stott subsequently informed me that 

 this second method had been originally discovered by that 

 redoubtable " philomath " Michael Dary, a contemporary 



1 Written in connection with Mr. W. Stott's paper. 



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