Arithmetic  of  impofjible  Quantities.  333 
cmxy—acmxyx—cm  aXxx J'cayojx.  The  fluent  of  the  firft 
member  of  this  equation  is  evidently  of  the  form  d cmxy, 
the  fluxion  of  which,  viz.  vcmxy+‘Dmcmxyx  being  com- 
pared with  the  former  gives  D=r, and  m=-a;  wherefore, 
c~axy-  J' ?~2axx J'caxo^x,  or  y=caxx  J'l :~2axx J' cax  <yx. 
Let  caxQj.i-%)  and  cr~2axx=  v;  then  j'c — 2axx J'cax q x - 
f zv-zv-J' 1 oz;  but  v-  ~ - fuppofing  that  v and 
x vanifh  at  the  fame  time;  therefore  vz  -fvz= 
~fcaxQjc-  ~fcaxQJx-  rJc^x^Jc-^x  = 
I f*  c — zax  f*  cax  f* 
Taj  c~ax  Q-*  - TtJ  caX(±A'  Hence  y = -J  c~ax  q>-  - 
£—ax  f* 
— J ca*oJc.  This  value  of  y is  fufficient  for  the  con- 
ftrudtion  of  the  fluent,  becaufe  the  quantities Jc~axoxi 
and \fcoJc  depend  on  the  quadrature  of  the  hyper- 
bola; but  if  we  would  introduce  into  it  the  ordinates 
and  abfcifles  of  that  curve,  we  need  only  have  recourfe 
to  the  foregoing  lemma,  from  which  it  appears,  that 
y—~a  ord.  ax J'ojh  abf.  ax-  ~a  abf.  ax J' \x  ord.  ax. 
13.  Let  the  co-efiicient  ofjy  be  now  fuppofed  affirma- 
tive, or  let  ^-+<2*^=q.  In  this  cafe  imaginary  expreffions 
are  introduced  into  the  fluent,  and  the  conftrudion  by 
U u 2 the 
