338  Mr.  PLAYFAIR  on  the 
Thefe  feries  differ  from  the  former  only  in  the  figns, 
and  the  arrangement  of  the  terms;  and  when  either 
or  n- 1,  is  di visible  by  4,  the  figns  remain  the  fame  in 
both. 
16.  The  reafon  of  the  foregoing  rule  foi  changing 
the  figns  is,  that  the  redangle  under  two  ordinates  to 
the  hyperbola  is  always  expreffed  by  the  difference  of 
two  abfciffie : and  that  if  from  the  abfcifs  belonging  to  a 
greater  fedor,  be  fubtraded  the  abfcifs  belonging  to  a 
lefs,  the  remainder  will  be  affirmative;  whereas,  if  from 
the  cofine  of  a greater  arch  be  fubtraded  the  cofine  of  a 
lefs,  the  remainder -will  be  negative.  Therefore,  that  the 
redangles,  expreffed  by  thefe  remainders,  may  have  the 
fame  fign,  in  both  cafes,  the  figns  of  the  remainders  muff 
the  different. 
It  appears  then,  that  the  fecond  rule,  as  well  as  the 
firft,  is  foiuided  on  the  principle  of  analogy  when  taken 
with  the  neceffary  limitations,  and  it  is  likewife  evident 
from  the  inftances  which  h^ve  been  produced,  that  thofe 
rules  lead  to  the  very  fame  conclulions  which  are  ob- 
tained from  the  imaginary  values  of  the  fine  and  cofine. 
There  are,  however,  inftances  in  which  the  analogy 
between  the  circular  and  hyperbolic  areas  being  wholly 
interrupted,  neither  the  foregoing  rules,  nor  any  of  the 
Tame  kind,  can  be  applied;  but  this  occafions  no  ambi- 
guity, 
