[ 38o  ] 
XVIII.  Qbfervations  on  the  Limits  of  Algebraical  Equa- 
tions'.; and  a general  Demonjlration  of  Des  Cartes’s  Rule 
for  finding  their  Number  of  affirmative  and  negative 
Roots.  By  the  Rev.  Ifaac  Milner,  M.  A.  Fellow  of 
Queen’s  College,  Cambridge.  Communicated  by  An- 
thony Shepherd,  D.  D.  F.  R.  S.  and  Plumian  Profeffior 
at  Cambridge. 
Read  February  26,  1777. 
§ 1.  **  I ^ HE  inveftigations  of  the  limits  of  equations  is 
“*■  confidered  as  one  of  the  moft  important  pro- 
blems in  algebra.  The  knowledge  of  them  not  only 
enables  us  to  dernonftrate  many  ufeful  theorems  in  that 
fcience,  but  is  alfo  of  material  fervice  in  difcovering  the 
roots  themfelves.  Mr.  maclaurin  has  treated  this  fub- 
je£l  very  fully,  both  in  his  Algebra  and  in  the  Philo- 
fophical  Transactions. 
The  fubftance  of  what  he  has  delivered  may  be  briefly 
exprefled  in  the  two  following  propofitions. 
1 ft.  That  any  equation  xn-pxn~ l+qxn~z-Ztc.  = 0 
being  propofed,  if  you  take  the  fluxion  of  this  equation, 
and 
