336  Mr.  PLAYFAIR  on  th& 
abfciffas  and  their  produdts  enter  into  the  enunciation, 
and  converfely.  Thus  abf.  a x abf.  b — — 1 •'  ~ - + abf'  “~b  - 
and  cof.  a x cof.  b~  + c—-a~h . The  fame  holds 
when  the  limple  power  of  the  ordinate  is  combined  with 
any  power  whatever  of  the  abfcifs : fo  in  the  theorems 
r , 1 1 r 7 ofd.  a-\-b  ord.  a — b , 
or  art.  5.  and  4.  ord.  a x abl.  b=  — ~ — ; and 
~ r , fin.  a-\-b  fin.  a — b 
lm.  a x col.  b — 4-  
2 2 
2.  When  an  expreflion  containing  any  property  of 
hyperbolic  fedtors,  involves  in  it  the  rectangle  of  two-or- 
dinates, the  value  of  that  rectangle  muft  have  a con- 
trary fign,  when  a tranfition  is  made  to  the  circle.  Thus 
ord.  1 a x ord.  ib  — * ; but  fin.  a x fin.  b = 
— c—“+b  + • The  difference  which  according  to 
this  rule  is  found  between  the  powers  of  ordinates  and 
of  lines  may  be  feen  in  the  following  examples.  If 
denote  any  hyperbolic  fedtor,  then,  by  involving  c—~  > 
and  again  fubftituting  for  the  exponential  quantities  as 
in  art.  5.  we  have, 
