Mr.  playfair  on  the 
33* 
J'pc~x^~'1.  Wherefore,  by  collecting  the  fum  of 
all  the  terms,  we  have  fin.  x J'pcoLx-coLx J'p  fin.  x - 
The  demonftration  in  the  cafe  of  the  hyperbola  is  free 
from  imaginary  expreffions;  but,  in  other  refpe&s,  is 
exactly  fimilar  to  that  which  has  now  been  given  in  the 
cafe  of  the  circle. 
1 2.  Let  the  co-efficient  of  y in  the  propofed  equation 
be  firft  fuppofed  negative,  that  is,  let  j,  —axy  — cl,  and  if 
we  multiply  by  cnxx,  n being  a conftant  but  indeter- 
minate  quantity,  it  becomes  -arcnxyx  = cnxoJc.  Let 
cnx  x - By  be  affumed  for  the  fluent,  a and  b being 
indeterminate,  and  let  its  fluxion  be  taken,  then, 
+ n Acnxy  - n%  cnxy  x = cn*  Qji. 
- b cnxy. 
Hence,  by  comparing  the  terms,  we  get  a=i,/za-b  = 0, 
n~B=a*;  therefore,  n=±a,  and  B -±.a\  for  n and  b let 
the  value  + a be  fubftituted,  and  for  a,  its  value, 
unity;  and  the  affumed  equation  becomes 4 -ay* cax~ 
J' :axojx,  or  4 -ay—c~axJ' caxQjx,  Let  this  equation  be 
multiplied  by  cmxx,  m being  indeterminate  as  before,  and 
cmxy 
