328  Mr.  Playfair  on  the 
—I  _J_  £‘=~a — mx  X \/——I  _J—  q — a-\-x  X \J — I — • c ~ ~a  ntx-\~x  X “~I 
I cx*J — I c — -f-  I 9 
in  which  expreffion,  if  the  lines  he  fubftituted  for  their 
imaginary  values,  we  have 
fin.  a — firux?  -f  mx — fin/a  — jV  fim  a 4-  tn  v - — vV  ^ 
2X1  — col'.  A* 
; fin . a+ fin . a +x  + Tin . a + 2X {in'),  E.  i. 
When  ab  — rc,  or  a—x,  the  propofed  feries  becomes 
fm.  x + iin.  2 x + lin.  3 x (»z)>  and  its  value  = 
Iin.  x — fin.  m- f- 1 x x-f- fin,  mx 
2 X 1 — col.  x , 
In  like  manner  it  will  be  found,  that  the  fum  of  the 
* colines  of  the  fame  arches,  or  cof.  a + cof.  a + x + cof.  a + ix 
cof.  a — cof.  a -j-  mx  — cof.  a — x cof.  a- f-  mx  — x 
2X1-  Cof,  X 
and  when  a — x,  cof.  x + cof  2 x + cof  3 a; ( m ) = 
cof.  mx  — cof.  m -f- 1 X x j 
2X1 — cof.w  z 
9.  To  folve  the  fame  problem,  in  the  cafe  of  the  hy- 
perbola, we  muft  follow  the  fteps  which  have  been  traced 
out  by  tbefe  imaginary  operations.  Let  abe  be  an 
equilateral  hyperbola  (fig.  4.)  of  which  the  center  is  f, 
and  the  tranfverfe  axis  af=i;  let  abf,  acf,  adf,  8cc. 
be  any  fedtors  In  arithmetical  progreflion,  and  let  m be 
their  number;  it  is  required  to  find  the  fum  of  all  the 
ordinates  bg,  ch,  dk,  8cc.  belonging  thofe  fedtors.  Let 
the  fedtor  afb  = y#,  and  the  fedtor  bfc,  which  is  the 
2 
common 
