OPERATIVE DIVISION 225 



done) and for obtaining Cardan's solution, and where " superior 

 division " and " synthetic operative division " are also referred 

 to, and the use of the Rule for solving differential equations 

 touched upon. 



Of course the results, both of numerical and of operative 

 division, are nothing but absolute identities if the remainder 

 is not neglected. Thus, in one case, 



c <f> + - c $ - C (j> 



9 9 



and so on. And the algebraic ratio -7 may be treated in pre- 

 cisely the same way. 



III. It is, of course, well known that the invert of a rational 

 integral operation of a degree higher than the fourth cannot 

 be expressed in finite algebraic terms ; but by means of opera- 

 tive division we can always find an algebraic expression in the 

 form of a convergent infinite series for each separate real root, 

 or sometimes for two or more roots, or for all the roots to- 

 gether — though I have not yet ascertained how actually to 

 calculate more than one or two real roots from the last kind of 

 series . 



The process by which this is done is very simple. 



Let [</>] x = y. 



Then = : [fl x = = : y, 



9 9 



o 

 and x =-r.y. 



9 



We then find the form of 2 by operative division, apply it 



9 

 to y, and equate the result to x. 



Now most equations [/] x = may be put in the form 

 [<f>]x — y in many different ways which may be called settings ; 

 and many different settings will yield different series. The 

 questions are, to which root or roots does each series apply ; 

 when is it convergent ; and which series gives the quickest 

 approximation ? 



In this part of my paper I will deal only with the applica- 



