Arithmetic  of  impojjible  Quantities.  331 
that  the  exponential  value  of  the  hyperbolic  ordinate 
may  be  deduced  from  what  has  been  proved  in  this 
article. 
1 1 . But  as  the  arithmetic  of  impoffible  quantities  is 
no  where  of  greater  ufe  tjian  in  the  inveftigation  of 
fluents,  it  is  of  confequence  to  inquire,  whether  the  pre- 
ceding theory  extends  alfo  to  that  application  of  it. 
% 
Let  it  then  be  required  to  find  the  fluent  of  the  equa- 
tion ^ zpay=Q^,  where  (^denotes  any  function  whatever 
of  x.  For  this  purpofe,  the  following  lemma  is  premifed : 
let  x be  any  arch,  and  p any  flowing  quantity;  then,  if 
the  fign be  taken  to  denote  the  fluent  of  the  quantity 
to  which  it  is  prefixed,  fin.  x J'p  cof.  a? -cof.  x fp  fin.  x=- 
-J  pc  xv  J—  2—~—J  pcXs/~l ; or  if  \x  be  a hy- 
av' 
perbolic  fedlor,  ord.  x Jp  abf.  x - abf.  x J'p  ord.  x — 
^ r ,,  P-  r CXV— ' — r-V— : > p.  (V-i+r-V— ' 
Became  fin.  xj pcoLx= — J px  — — 5 
by  feparating  the  terms  we  have  fin.  xfp  cof.  x — 
CX\/ — r f*.  , , i — *\/- 
1 
1 
4V 
i fpc *v'~I 
\f  *a*~  I 
— I 
■fp 
£ A V - I 
'“y~ Sfpr  the  fame  reafon  - cof.  xj'p  fin.  x— 
-c^fpcXV~l  + 
U u 
4v'  — ii/  ■*  ' ' 4V — 1 
Vol.  LXVIII. 
r~*V — 1 
4V— i 
