OPERATIVE DIVISION 411 



since it is only at these roots that f'X is negative. At the roots 

 X 2t Xt, X 6 . . ., f'X is positive ; therefore 1 + f'X is greater 



than unity ; therefore -y- [0 + f]X is greater than unity ; there- 

 fore [0 4- f]X is infinitely large when m is so; therefore 



[0 + /]* (X + h) = X only when h = o. 



(5) We find then the remarkable fact that if/', or a vicarious 

 curve, lies within certain limits, the curve [0 +/] n approxi- 

 mates when n is indefinitely increased to a series of finite 

 straight lines alternately parallel and perpendicular to the axis 

 of x ; the parallel straight lines passing successively through 

 the points (X lt Xi,), (X 3> X z ), (X 5 , X 6 ) . . ., and the perpendicular 

 lines through the points (X 2 , X 2 ), (X^X^) . . . ; the lines meeting 

 end to end ; and the lengths of the inner lines parallel to x 

 being equal to the differences between the even roots, and those 

 of the inner perpendicular lines being equal to the differences 

 between the odd roots — in fact, like the section of a flight of 

 steps. Or, perhaps, we ought to consider that the perpendicular 

 lines do not exist at all, and that the curve [0 +/] ° becomes 

 discontinuous at the even roots of fx = o. 



The reader will have no difficulty in constructing geometric 

 illustrations of these propositions, especially by the use of the 

 operative-unit curve, or mid-axis, 0. The equation o = o*6 — 

 i* 1 x + o'6 x % — x\ of which the roots are 1, 2, 3, affords a 

 convenient example ; but he should observe that the proposi* 

 tions apply to many equations besides rational integral ones. 

 For the latter the results may be summarised as follows : 



If AT is a root of the equations o = / (x) and o = F (x), and 

 if F and x are properly selected, then 



n=o 



2 [F][o+Ffxo~[o +F] M x Q 

 F(o). 



F(o) - F(x) 



Further proofs with geometric illustrations, and with corol- 

 laries and applications, will, I hope, be given in the third Part. 1 



1 The validity of solution by iteration is usually based upon the familiar theorem 

 that if f n {x) comes to a limit that limit is a root of the equation x =f(x). But this 

 is of itself insufficient because there is no a priori reason whyy~ n (;r) should come to 

 any limit at all, and secondly because the usual proof of the theorem referred to, 

 though easily obtained, does not indicate the conditions of convergence. 



