and  c  —  0,  the  expression  (62  —  4ac)/+  ae2  4  cd2  -  bed  reduces
to  b  (bf—  de}\  thus  we  conclude  that  when  Z>2  —  4ac  is  positive
the  equation  (1)  always  represents  an  hyperbola,  except  when
(62  —  4ac)/+ae2  +  cd*  —  bed  =  0,  and  then  it  represents  two
intersecting  straight  lines.
III.     Suppose  b*  —  4ac  =  0,  then  (2)  becomes

±    f2  (be  ~


which  may  be  written  y  =  a#  -f  ft  ±  —  (p'#  +


where


2  (6e  -  2ccZ),    q'  =  ez-  4c/.
If  ^)'  be  positive,  the  expression  under  the  radical  is
positive  or  negative,  according  as  a?  is  algebraically  greater

or  less  than  —  ^  ;  if  ^'  be  negative,  the  statement  must  be

reversed.  In  both  cases  the  curve  extends  to  infinity  in  one
direction  only  and  is  therefore  a  parabola.
The  straight  line  y  =  ax  +  /3  is  a  diameter,  bisecting  all
ordinates  parallel  to  the  axis  of  y,  and  meeting  the  parabola

at  the  point  for  which  x  =  —  —..
P

If  p  =  0,  the  equation  becomes  y  =  ax  +  /3  +  -~-  ;    this

equation  represents  two  parallel  straight  lines  if  q  is  positive,
and  one  straight  line  if  q  =  0  ;  if  q  is  negative,  the  locus  is
impossible.
