584 SCIENCE PROGRESS 



If X q is the root to which this series refers, then by Part II, 

 p. 397, descending operative division gives us 



^_-7i-{(ag-(a©>-— •• 



where g= V —p q /p for short. It is convenient to employ the 

 notation suggested in the same section, p. 398, according to 

 which yfr q x denotes a descending function of which the leading 

 and highest term contains x q ; and <f> q x denotes an ascending 

 function of which the leading and lowest term contains x q . 

 Then if 



f q x = - p q /p , X q = [f q ]~\ - p q /p Q ) ; 



with similar equations for <j> q . 



If p s , p t , . . . are other negative terms, we can also divide 

 fx by p Q x n ~ s , pQX n ~\ . . . and obtain similar inverts by descending 

 operative division for X s , X t , . . . where these roots may or 

 may not be the same as X q . Thus we may consider as many 

 descending inverts as there are negative terms in fx ; and 

 may observe that the subjects of all will be positive real 

 numbers. But if p 1 be a negative coefficient, we can obtain 

 yet another invert from fx/x 11 , namety [yjr_ i y i ( — p /px). On 

 examining this, however, we find that it is the same as 

 [ ,> / r i]~ 1 ( —pi/Po)> with a rearrangement of terms. 



By writing fx = o in an ascending series <f> x = o, putting 

 p n positive and dividing by it multiplied into the powers of 

 x contained in terms which are now negative, we obtain a set 

 of ascending inverts. If p ll _ 1 is now negative, we have got 

 another ascending series (without division of </> # bv any 

 power of x), namely [0i] _1 ( — p n /p n -\) — which is indeed the 

 series given by simple ascending division as shown in Part I, 

 Section III (3), p. 230. Here again, however, this proves to 

 be the same series as [<£_i] -1 ( — p n -i/Pn)- 



If, in any of the descending settings such as yjr q x = — p q /po 

 we put z' 1 for xi we obtain ascending settings in z which give 

 ascending inverts on the same subjects such as — p q /p . 

 These prove to be, by the use of the multinomial theorem, 

 nothing but the algebraic reciprocals of the descending inverts 

 n x — as was to be expected since the roots of z must be the 

 algebraic reciprocals of the roots of x. Compare also Part II, 

 Example H, p. 400, where we saw that </>:* . ty- 1 = 1 . Similarly 





