Mr.  milner’s  Qbjervations , See.  381 
and  divide  it  by  x,  the  refulting  equation  will  have  all  its 
roots  limits  of  the  roots  of  the  given  equation. 
2dly,  If  the  terms  of  the  propofed  equation  be  multi- 
plied into  the  terms  of  any  arithmetical  feries,  the  re- 
fulting equation  will  alfo  have  its  roots  limits  of  the  roots 
of  the  original  equation. 
§ 2.  This  fecond  proportion,  though  admitted  by  all 
the  eminent  authors  whom  I have  had  an  opportunity  of 
confulting,  certainly  requires  fome  reftriftions.  For 
example,  the  roots  of  the  quadratic  equation  x 2 x " 3 " ^ 
are  3,  See.  - 1 ; multiply  the  terms  of  this  equation  into 
the  terms  of  the  arithmetical  progreffion  1,  2,  3,  refpec- 
tively,  and  the  refulting  equation  is  1 xx1-  2 x 2^-3  x 3=0, 
the  roots  of  which  are  2±V  1 3,  neither  of  which  are  be- 
tween the  roots  of  the  given  quadratic. 
Again,  fuppofe  the  roots  of  the  cubic  equation 
xi-px2+qx-r=o  to  be  a,  b,  -c,  and  it  is  poffible  that  the 
equation  1+  $mxxi—l+2mxpx1‘  Jrl+mxqx-lr-°  may 
have  no  root  between  the  quantities  b and  -c;  and  in 
general,  if  the  roots  of  the  equation  (a)  xn~pxn'~1+qxn~~2, 
See.  — 0 be  fuppofed  a , b,  c , -d,  -0,  -/,  &c.  where  a is  the 
greateft  root,  b the  next,  and  fo  on  in  order,  the  equation 
(p)l+nm-x.xn-l+n-i.mpxn—ljrl+n-2.mqxn~2i  Sc c.  = 0 
will  not  necelfarily  have  any  of  its  roots  between  the 
roots  c and  -d  of  the  original  equation. 
C c c 2 § 3* 
