Arithmetic  of  mpofible  Quantities. 
329-, 
comnaon  difference  of  the  fe£tors,  =~x:  then  bg,  or 
ord.  a — 
5* 
Ca 
ca+x. 
, by  art/ 
, and  ch,  or  ord.  a+x- 
2 7 7 2 
Therefore  the  feries  of  ordinates,  that  is, 
BG+CH+DK+ 
c“ 
2 
(m) 
= T* 
X I + C~X  + C~2X . . .....  (m)  — . — X 
Ca ca  + mx Ca — x -}-  ca  + mx — x — - 0~a  -j- 
I c — mx 
Cx I 
ord.  a — ord  a -f  mx  ord.  a — x -f  ord^-tf  + mx  — x 
When  a — Xy 
2X1  — abi.  X 
ord.  x + ord.  zx  + ord.  z,x  + ....  . (m)  = 
ord.  x— ord.  m + i X x-f  ord.  mx 
1 1 " — — i—  » 
2X  i — abf.  x 
In  like  manner  it  is  proved,  that  the  fum  of  the 
abfciffae,  that  is,  fg+fh+fk+  .....  (m)  = 
abf  a — -abf  a-\-mx  — abf  a — #-}-abf.  a-\-mx — xm 
2 x i — abb  x 
; and  when  a=x,  this 
sv-%  -»  abf.  mx- t-  abf.  m -f  i X x t 
expreffion  becomes •••. 
io.  The  coincidence  of  the  theorems  deduced  in  the  : 
two  laft  articles  is  obvious  at  firft  fight,  and  if  the  me-, 
thods  by  which  they  have  been  obtained  be  compared,  . 
it  will  appear,  that  the  imaginary  operations  in  the  one 
cafe  were  of  no  ufe  but  as  they  adumbrated  the  real  de- 
mon  ft  ration,  which  took  place  in  the  other.  This  will 
be  rendered  more  evident  by  confidering  that  the  refolu- 
tion  of  the  feries  of  hyperbolic  ordinates,  into  two  others  ; 
of  f 
