OPERATIVE DIVISION 587 



It will be seen that this is generating the invert given in Part 

 II, p. 397, namely 



WW" 1 * -* + 36 + (\c + l f} g -i + (hh + '-be + ±bty-* + 



+ -^~ 3 + etc. 

 3 



= *i -Mo + '-1 + *-a + '-3 + • • • (say). 



Thus Ig gives t x + tf correctly ; Pg adds /_! correctly ; Pg 

 adds /_ 2 ; and so on. The work will be much facilitated by 

 Table I, which gives the first few terms of the ordinary multi- 

 nomial expansion of (yjr n ) and (<£ re ) r . 



We may verify this result by the second method mentioned 

 above, according to which I operating once on an invert 

 generated by it reproduces the same invert ; and the work will 

 be facilitated by the formula for (^,r 1 ) r in Part II, p. 399. 

 This is what was to be expected. For if the invert rightly 

 expresses the root, whichever root it may be, then the result 

 when / operates upon it should be zero. In fact we shall 

 find, not only that [^ g + p q ] fy q ]~ l (— p q ) = o, as indicated in 

 Part II, Example I, p. 400, but that more generally 



Wv+/v]tyg- 1 (-/>,)=o, 



if p Q = 1 . This holds when r = n ; so that I = when it 

 operates upon any of the appropriate inverts. 



(4) But many other iterands will have the same result ; 

 and it is necessary to mention them in order to show that 

 all the forms of iteration described in Section VI (3, 4, 5) will 

 generate the same series if the base and modifier be properly 

 chosen. For axial iteration, the general theorem is that 



^ r -g r 



1= 0- 



qg r ' 1 



operating on g will generate [y^ 5 ] -1 ^. If r= n and #=3, 

 this is the example just examined. Or we may take r= q. 

 Whatever the value of r, Ig will be the same as when r— n, 

 and Pg will give the series right to about five major terms. 

 But unless r = n, ^r r — g r will contain negative powers of 

 which will require irrational expansions of the subject. 



For Newton's form of axial iteration take 7 = — (yjr q — g q )/-*jr' q 

 and iterate on g as before. Thus for the example used above, 

 on expanding the fraction by algebraic division, we have 



