Limits  of  Algebraical  Equations,  &c.  383 
negative  one;  and  then  the  four  refults  will  be  -m~R.af 
+msb,  —m t c,  -mzd. 
§ 4.  In  general,  the  roots  of  the  equation 
nxn~l-n- x . pxn^2+n— 3 . q xn~3,  are  always  between 
the  roots  of  the  equation  (a)  becaufe  the  roots 
of  this  laft  equation'  fubftituted  fucceffively  for  x in 
nxn~l-n- 1 . pxn~ 2 + &c.  always  give  the  refulting 
quantities  alternately  negative  and  pofitive;  but  when 
the  leaft  of  the  affirmative  roots,  and  the  greateft  of  the 
negative  roots  of  the  equation  (a)  are  fubftituted  in  (b)  the 
quantities  that  refult  will  neceffarily  have  the  fame  fign, 
and  therefore  it  is  poffible,  that  no  root  of  the  equation 
(b)  may  lie  between  the  lead  of  the  affirmative  and  the 
greateft  of  the  negative  roots  of  the  equation  (a). 
§ 5.  It  is  poffible  even,  that  the  equation  (b)  may  have 
imaginary  roots,  at  the  fame  time  that  all  the  roots  of 
the  equation  a are  real,  which  is  contrary  to  what  all  al- 
gebraical writers  have  thought.  For  inftance,  the  roots 
of  the  equation  x2  + 6 x—  7 —0  are  7 and  — 1,  and  if  the 
terms  of  this  equation  be  multiplied  by  1,  — 1,  3 (an 
arithmetical  feries  where  the  common  difference  of 
the  terms  is  equal  to  2)  the  refulting  equation  will  be 
x1- 6 x4  a.  t , the  roots  of  which 'are  evidently  impoffible. 
§6.  However,  theequation(B)  caii  never  havemore  than 
two  imaginary  roots,  when  the  roots  of  the  equation  (a) 
lfi'  7 
are 
