Making  this  substitution,  the  equation  becomes

mx'''-\-  2lx'y'  +  ny'''  +  2(a'm  +  h'l  +  p)x'^  2(a'l  +  b'n  +  q)y'
+  a''7n  +  b''n  +  2a'b'l  +  2a'p  -f  'Z^q  +  c?  =  0.         (2)

We  now  so  determine  the  co-ordinates  a'  and  b'  that  the
coefficients  of  x^  and  y'  shall  vanish.     To  effect  this  we  have

the  equations

nm'  +  lb'  =-p',)  (^)

la'  +  nb'  =  -q;)  ^  ^

in  which  «'  and  b^  are  the  unknown  quantities.     Solving  the

equations,  we  find

.  _np-lq^^

^        72  >      I

I  -mn   [  .  (3)

r  —  mn  J

Omitting  for  the  present  the  special  case  in  which  ?  —  mn
=  0,  these  values  of  a'  and  b'  will  always  be  finite.
By  means  of  these  values  of  a'  and  b'  we  may  simplify  tlie
equation  (2)  as  follows:  Multiplying  the  first  of  equations  (a)
by  a',  the  second  by  b\  and  adding,  we  have

ma"  +  nb"  +  2Ia'b'  =  -  a'p  -  b'q.  (b)

By  means  of  the  equations  (a)  and  (J)  the  general  equation
(2),  omitting  accents,  is  reduced  to

mx^  +  2lxy  +  nf  +  a'p  -^b'q^d^^,  (4)

This  equation  (4)  will  now  represent  the  same  curve  as
(1),  only  referred  to  new  axes  of  co-ordinates.
