Limits  of  Algebraical  Equations,  Sec.  387 
any  equation  is  always  the  product  of  all  the  roots  with 
their  ligns  changed,  the  number  of  pofxtive  roots  in  each 
of  the  equations  (d)  and  (e)  muft  be  even : therefore,  the 
number  of  politive  roots  in  (d)  cannot  exceed  the  number 
of  thofe  in  (e)  by  unity ; but  there  is  in  (d)  one  root  more 
than  in  (e),  and  confequently  it  muft  be  negative. 
If  both  the  terms  l and  mx  are  negative,  becaufe  then 
the  number  of  politive  roots  in  (e)  and  (d)  are  even,  it 
follows  in  the  fame  way,  that  there  is  one  negative  root 
more  in  (d)  than  there  is  in  (e). 
And  laftly,if  the  terms  l and  mx  have  different  ligns, 
for  the  fame  reafons  there  muft  be  one  politive  root  more 
in  the  equation  (d)  than  there  is  in  (e). 
des  cartes’  rule  is,  that  there  are  as  many  politive 
roots  in  any  equation  as  there  are  changes  in  the  ligns  of 
the  terms  from  + to  — , or  from  - to  +,  and  that  the  re- 
maining roots  are  negative.  From  what  has  been  de- 
monftrated  it  appears,  that  if  this  rule  be  true  in  the 
equation  (e),  it  muft  hold  alfo  in  the  next  equation  (d)  of 
fuperior  dimenlions ; and  as  we  know  that  it  is  true  in 
limple  and  quadratic  equations,  it  muft  therefore  be  true 
in  cubics,  in  biquadratics,  and  fo  on. 
This  is  one  of  the  heft  rules  we  have  in  algebra.  Dr. 
SAUNDERSON  <e>  faw  fuch  an  infinity  of  cafes  in  equa- 
(c)  Vol.  II*  p.  683.  Algebra. 
Vol.  LXVIIL  D d d 
tions 
