OPERATIVE DIVISION 409 



Obviously, so far as we have examined the iterants, the two 

 results agree, since the same groups of the original coefficients 



occur in both and because the powers of j- in the quotient 



are probably the sums of the powers of b within the small 

 brackets in the iterants (<V) when continued to infinity. 

 But we have still to explain why the powers of x in the iteration 

 series do not appear in the quotient. 



For descending division take the form used in the preceding 

 subsection to illustrate Dary's method, namely [V5 4- 2o].r . 

 Expanding this by the binomial theorem we obtain 



Xi = g + -g- 2 . 2Xq £~ 5 .4# 2 + . . . 



where g = 4/5. This is the "general case" just briefly dealt 

 with ; and it will be easily seen that x 3 is taking the value of 

 the operative invert given in Section IV (1), Example A. This 

 suggests that the descending inverts represent the iteration, 

 not of the original operation, but of vicarious operations. 



Perhaps these conclusions were to be expected ; but now 

 consider Newton's vicarious form in the case of o = a — x -J- x 2 ; 



that is, iterate -\ — — . By simple algebraic division 



1—20 



this becomes 



Xi = a + 2clxq 4- (4<2 — i)#o 2 + (4a — 1)2*0 3 + • . . 

 This is the " general case " again. Iterating directly (and 

 easily if x = o) or by the formula for 4> 3 just given, we have 



x 3 = a + « 2 4- 2& 3 4- 5« 4 + M« 5 + 42a 6 ... 4- Rx . 

 This then is precisely the same series as the one first obtained by 

 simple iteration. Observe however that in Newton's form x 3 

 approximates as closely as x 6 did by simple iteration ; and 

 note that the operative solution therefore gives the quickest one. 



If, then, this theorem is generally true we shall have ob- 

 tained two important results. First, operative division enables 

 us to give, at least in many cases, an algebraic expression to 

 the arithmetic process of iteration ; and, secondly and con- 

 versely, iteration often enables us easily to determine the con- 

 ditions of convergence in the operative series. 



But we have still to explain the disappearance of the terms 

 in x Q . 



(4) I have space in this Part to select only an incomplete 

 27 



