5 82 SCIENCE PROGRESS 



If only the last and absolute term be negative all the critical 

 points refer to the only root but are all retrospective — so 

 that the root is less than the least of them. In ^+ ioo#* 

 -\- x— 1 =o, the root ( = 0-095 • • •) 1S ^ ess tnan °* l • By putting 

 i/z for x we transform the first of these cases into the second 

 and vice versa, but get no new information. 



Where / consists of a set of positive terms followed by a set of 

 negative ones, the root lies between the greatest prospective 

 and the least retrospective critical point. Thus in 



X s + 3* 2 — 2x — 5 = o ; 

 3/3+ Ss/3 + 



the root (= 1*33 . . . ) lies between «/5/3 ■+- an d -/2— . 

 (Observe the method used for denoting prospective and retro- 

 spective critical points.) 



For two changes of sign, consider 



x? — 3# 3 — 2x-\- 5 = o. 

 3 + ^2 + 



vV3- 

 5/2- 



The critical points under negative terms refer to the odd and 

 greater root, and those under the positive term to the even 

 root — if these really exist. Thus the former (= 3*128...) 

 is greater than 3, and the latter ( = 1-201 ...) is less than 

 1-29 . . . — see Part I, p. 232, and Part II, p. 395, and note 

 that the inverts are operations performed upon the appropriate 

 critical points. In the equation x 2 — x -\- a =0, dealt with 

 in Part II, p. 407, the even critical point a is the subject both 

 of the iteration and of the ascending operative invert — both of 

 which must therefore refer to the even and lesser root, counting 

 downwards. For the negative roots of Newton's equation 

 x? — 2x+ 5 = o, we find that the odd critical point is -J 2 — 

 and the even one is ^5 + ; that is the odd root should be less 

 than the even one — but this is impossible and both roots 

 must therefore be unreal. 



For three changes of sign consider x z — i6x 2 + 65*— 5 =°> 

 of which the roots are 1,5, and 10. The even critical point 

 65/16+ refers to the middle root ; the odd point 16— to the 



