Mr.  Playfair  on  the 
.324 
equations  a value  of  the  cofine  may  be  found  in  the  fame 
imaginary  terms  which  were  affigned  above.  Now  by 
means  of  thefe  expreffions  many  theorems  may  be  de- 
monftrated;  it  may,  for  example,  be  Ihewn,  that  if  a and 
b are  any  two  arches  of  a circle,  of  which  the  radius  ia 
. , ~ r , fin  a+b  fin.  a — b ^ - 
unity,  then  fin.  ax  col.  b — — - — + — — — . r or  lin.  a— 
* yj/ZTT 9 and  cof.  b=  — — — therefore,  fin.  a 
ca+bXis/—  I c — a~~b X \Z~~ •!  -f  C 'a — V'— 1 c&-~> «X\/ — I 
X cof.  b = ^7=1 = 
fin.  «-{-£  fin.  # — £ 
-r 1 
2 2 
5.  Now  it  may  be  obferved,  that  the  imaginary  value 
which  has'been  found  for  fin.  a was  obtained  by  bring- 
ing a fluxion,  properly  belonging  to  the  circle,  under  the 
form  of  one  belonging  to  the  hyperbola.  It  may,  there- 
fore, be  worth  while  to  inquire,  whether  a fimilar  ex- 
preflion  may  not  be  derived  from  the  hyperbola  itfelf. 
Let  bad  be  a rectangular  hyperbola  (fig.  2.)  of  which 
the  center  is  c,  and  the  femi-tranverfe  axis  ac  = 1 ; let  b 
be  any  point  in  the  hyperbola,  join  bc,  and  let  be  be  an 
ordinate  to  the  tranfverfe  axis.  Then,  if  the  feCtor 
acb  and  be=js,  it  is  known  that  a— — ; whence 
s/l+z 2 
a = log.  & + V 1 +z%  and  ca=z+\/ 1 +zz.  But  if  the  feCtor 
be  taken  on  the  other  fide  of  the  tranfverfe  axis,  a and  z 
become 
