OPERATIVE DIVISION 585 



all the ascending inverts of the equation in x are the algebraic 

 reciprocals of the descending inverts of the equation in z. 



If p and p n have the same sign (so that there may be an 

 even number of real positive roots), then there will be as 

 many descending as ascending inverts, all with positive sub- 

 jects ; so that there will be twice as many inverts as the number 

 of negative terms in fx. If p Q and p n have different signs (so 

 that there must be an odd number of real positive roots), 

 the invert [-v/r„] _1 ( — p tl /po) will be found to be the same as 

 [$-■"] _1 ( ~ Po/Pn) ) so that the total number of inverts will 

 equal the total number of terms less one. 



(2) On examining the series in Part II, p. 397, we see 

 that each operative invert, descending or ascending, consists of 

 a series of positive or negative integral powers of operating 

 on the appropriate fractional power of the subject. Indeed, 

 the remarks under Example F, p. 399, show that 



M _1 ( - PJPo) = iWJr 1 V - p q /Po ; 



the operation [y^J -1 being the series of integral powers 

 of just mentioned. Now v 7 — pjpo is a critical point as 

 defined in the previous section. Thus 



Each invert obtained by operative division consists of an 

 operation performed upon a critical point. 



Hence we infer that each invert refers to the same root of the 

 equation as the one to which the critical point refers. 



(3) Next it remains to be shown that each invert obtained 

 by operative division is the algebraic expression of various infinite 

 iterations based upon the same critical point. 



Three special examples of this were given in Part II, 

 Section V (3) — in two of which the same invert was proved to 

 be generated by two different iterands. We now examine 

 the general case. 



First observe that in the equation yfr q x = — p q /po we have 

 hitherto brought the whole of the absolute term to one side of 

 the equation, so that yfr q does not contain one. But we may 

 if we please add any arbitrary number to both sides of the 

 equation. We may call this a modifier and denote it by ^ m : 

 so that if yjr q x is now supposed to contain m, the equation may 

 be written 



y\r q x = m — pJpQ = y (say). 



