384  Afr.  milner’s  Obfervations  on  the 
are  real.  For  fuppofe  thefe  laft  roots  to  be  +a,  +b,  +c, 
+d,  -e,  -/,  See.  in  their  order  from  the  greateft  to  the 
leaft,  and  fxnce  the  refults  which  arife  from  the  fucceffive 
fubftitution  of  thefe  quantities  are  always  alternately 
negative  and  pofitive,  that  cafe  only  excepted  where  d and 
-e  are  fubftituted,  it  is  manifeft,  that  we  fhall  always 
have  n—2  of  the  roots  of  the  equation  (b)  which  will  be 
limits  of  the  equation  (a). 
§ 7.  It  is  remarkable,  that  whenever  the  equation  a 
has  all  its  terms  complete,  its  roots  real,  and  fome  of  them 
pofitive,  and  others  negative,  if  l+nm  be  aflumed  equal 
to  0,  the  equation  b will  always  have  one  of  its  roots 
either  greater  than  the  greateft  affirmative  root,  or  lefs 
than  the  leaft  negative  root  of  the  equation  (a).  Thus, 
in  the  quadratic  x1  + 6x-j=o,  affume  any  arithmetical 
progreflion  o,  1,  2,  the  firft  term  of  which  is  equal  to 
nothing,  and  the  equation  b in  this  cafe  is  6x- 14=0  and 
x =qr , which  is  greater  than  1,  the  greateft  affirmative 
root  of  the  aflumed  equation. 
§ 8.  The  roots  of  the  equation  (a)  being  ftill  fuppofed 
a,  by  c,  d,  —et  -/,  See.  let  m be  taken  equal  to  unity,  and 
/ any  pofitive  integer  whatfoever,  and  in  that  cafe,  two  of 
the  roots  of  the  equation  B will  lie  between  the  roots 
d and  and  one  of  them  will  be  pofitive,  and  the  other 
negative. 
For 
