Arithmetic  of  impojibk  Quantities.  341 
cof.  a — cof,  #~~*eof.  T 
*"Ax  * 5T-— cof.  x ~ — o • In  like  mnnner  do 
the  fums  of  all  the  fubfequent  columns  vanilh ; and 
therefore,  cho.  «+cho.  tz+v+cho.  a+  2X  ....  . ( m)-mp . 
-But  when  n is  an  even  number,  x l2if x — 
,r  i»— 1 2 x 
==  -T^^TTrr:^-  x If  therefore  the  radius  be  put 
= r,  and  the  expreffion  made  homogeneous,  we  have 
— » « « 1 2 t 7 n 1 
*FA  +FB  +FC  (ni)  zz  ftl  x — r X 23,t". 
^ ' I-2-3-4 f« 
E.  I. 
This  laft  coincides  with  the  forty-firft  of  the  curious 
and  difficult  propolitions  publifhed  by  Dr.  stewart, 
tinder  the  title  of  general  theorems : it  is  given  there 
without  a demonftration,  but  appears  plainly  to  have 
been  inveftigated,  in  a manner  altogether  rigorous,  by 
that  profound  geometer.  It  may  therefore  be  regarded 
as  one  of  the  inftances,  in  which  the  conclufions  of  this 
imaginary  arithmetic  are  verified  by  the  geometrical 
analyfis. 
17.  The  two  foregoing  propofitions  being  confined  to 
the  circle,  and  yet  having  been  inveftigated  by  the  help  of 
imaginary  expreffions,  may,  at  firft  fight,  feem  excep- 
tions to  the  rule,  which  we  have  been  endeavouring  to 
eftablifh.  But  it  needs  only  to  be  remarked,  that  they 
are  particular  cafes  of  certain  theorems  belonging  both 
X x 2 to 
