OPERATIVE DIVISION 589 



\3 9 81 / 



(5) For ascending division, write the original equation 



<f>_ n X = />.#- + p n .,X- n + l + p n -2pC~ n+2 + • • • + />0 = 0. 



Dividing by p n x~ n+q , and adding a modifier, we obtain 



<£_ g # = m - />,V/>» = y ; X q = [<f>_ q y i y. 

 For simple modified iteration, we find that, after dividing by p n , 



l=Q+4>-Jqg- n ~ l 



generates the series, where g~ q =y. This is the same as the 

 formula for generating descending inverts by simple iteration, 

 except that the signs of n and q are changed ; and the results 

 are also given by putting z~ x for x in the descending equation. 

 Both for ascending and for descending division, the cases 

 when q = o lie somewhat outside these formulae as they deal 

 with the inverts of § x and -ty_ x . But as stated in subsection 

 (1) above, these are the same as <£:} and ^f 1 (except that we 

 cannot use modifiers) ; and I have no space to discuss the 

 matter here. But simple ascending division as described in 

 Part I requires this setting and is so important that the gener- 

 ating iterands must be noted. Let 



<j) x == a — bx+ ex 2 — dx* -f . . . = o ; 



then it will easily be found that 



X= [o + <}>o/bYA= A + CA 2 + ( 2 C 2 -D)A Z + 

 + ( 5 0-sCD+E)A* + 



which is the operative invert when A. B, C, . . . = a/b, b/b, c/b . . . 

 Similarly for Newton's iterand — <£ /</>'o, we have by 

 ascending algebraic division 



1= A+ 2ACO+UAC 2 - 3AD- C)o 2 + (12 ACD+ SAO + 

 + 4AE - 2C 2 -2D)o 3 +(i6ACE+ 16AC*- 36AC 2 D + 

 + 9AD 2 - $AF+ 7CD- 4 C 3 -3E)o i + . . .; 

 IA=A+ CA 2 + (2C 2 - D)A" + (8C 3 - uCD+E)A*+ . . .; 

 PA = A + CA 2 + (2C 2 - D) A*+($C*-5CD+E)A*+ . . . 



The result will also be the same if we use only the first two 

 terms of <f>' , namely — b+ 2co — which is as it ought to be. 



