OPERATIVE DIVISION 581 



same sign as the inferior critical term as placed in the original 

 equation : otherwise it is retrospective. From (d) of the 

 previous subsection. 



(g) If the highest term of the equation be positive, all 

 positive critical points derived from pairs of which the inferior 

 terms are negative refer to odd positive actual roots, counting 

 from the greatest root downwards ; otherwise the positive 

 critical points refer to the even positive actual roots. This 

 is merely a restatement of proposition (c), because by Descartes' 

 Rule, if p and p n have the same sign there may be an even 

 number of positive actual roots, and if they have different 

 signs there must be an odd number of such roots. 



Thus if/ has only one change of sign, all the positive critical 

 points are derived from pairs with negative inferior terms 

 and must therefore refer each to an odd positive root ; that 

 is, they all refer to the same root — the first and only positive 

 root. If / has two changes of sign, the critical points from 

 pairs with negative inferior terms refer to the first and greater 

 root ; and the others refer to the second and lesser root. 

 If there are three changes of sign the critical points from pairs 

 with negative inferior terms must all refer either to the first 

 and greatest or to the third and least root ; and the others to the 

 middle root. And so on — equal roots being counted separately. 



Note that by (e) critical points which are greater than the 

 greatest root or less than the least root need not necessarily 

 refer to any root at all ; and that as roots become unreal in 

 pairs, their doing so does not affect the truth of the propositions 

 given above. 



(3) I have space only for a few examples. If only the 

 highest term of the equation be positive, all the critical points 

 refer to the only root and, by (/), are all prospective — so that 

 the root is greater than the greatest of them. Thus in 

 x 3 — 2x 2 — 2x — 3=0, the critical points are 2, %/2, and Z/3, 

 and the first of these is nearest the root (= 3). In x 3 — x 2 — 

 8 1 #—90= o the critical point 9 is nearest the root (=10). 

 In Newton's equation ^5 is nearest the root (=2-09...). 

 Observe therefore that the nearest critical point should be a 

 convenient base for iteration ; and, secondly, that in the case 

 of Newton's equation it is the subject of the invert given by 

 descending operative division — compare Part I, p. 232, and 

 Part II, pp. 394, 409. 



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