Litnits  of  Algebraical  Equations,  fee.  385 
For  example,  the  quadratic  equation  x2  + 6 x—  7 =0  has 
its  roots  x and  -7 ; and  if  the  terms  of  this  equation  he 
multiplied  into  3,  2,  1 ; 4,  3,  2 ; or  5,  4,  3,  fucceffively, 
the  refulting  quadratic  in  every  cafe  will  have  its  two 
roots  between  the  roots  of  the  given  equation,  and  one  of 
them  will  be  politive,  and  the  other  negative. 
§ 9.  The  equation  b,  which  in  the  laft  article  was  de- 
duced from  the  equation  a by  taking  m equal  to  1 , and 
/ any  politive  integer,  may  itfelf  be  treated  in  the  fame 
way,  and  the  refulting  equation  will,  a fortiori,  have  two 
of  its  roots  between  the  roots  d and  —e  of  the  original 
equation,  and  one  of  them  will  be  politive,  and  the  other 
negative. 
§ 10.  Let  x2-px+q-o  reprefent  any  quadratic  equa- 
tion, the  real  roots  of  which  are  a and  /3;  fuppofe 
and  we  lhall  have  1 -py+qy'1"=o,  the  roots  of  which  equa- 
tion are  £ , j.  Let  the  root  of  the  equation  2 qy-p-c 
be  equal  to-^-,  and—  will  always  lie  between  the  quan- 
tities , y,  and  therefore  one  would  think  at  firft  light 
that  the  quantity  a muft  always  lie  between  x and  (2. 
But  this  would  be  contrary  to  what  is  proved  in  art.  7. 
In  the  prelent  cafe  a can  never  lie  between  a.  and  /3,  un- 
lefs  thefe  two  quantities  have  the  fame  lign,  and  it  is 
obvious, 
