This  demonstration  is  however  not  complete.  For  in
Art.  394,  there  is  the  limitation  that  the  straight  line  which
is  projected  to  infinity  does  not  intersect  the  conic  section,  and
thus  Pascal's  Theorem  is  only  established  for  such  figures  as
conform  to  this  restriction.  Writers  who  use  the  method
of  projections  are  accustomed  to  assume  that  theorems  which


EXAMPLES.      CHAPTER   XVII.  333

are  demonstrated  under  certain  limitations  will  hold  when
those  limitations  are  removed.  For  example,  a  straight  line
which  really  does  not  meet  a  curve,  is  called  an  ideal  secant,
and  treated  in  effect  as  if  it  did  meet  the  curve.  But  it
would  be  beyond  the  scope  of  an  elementary  work  like
the  present  to  discuss  the  grounds  on  which  the  assumption
is  based.
EXAMPLES.
The  following  Examples  may  be  treated  by  the  method
of  projections:

•1.  CP  and  CD  are  any  two  conjugate  semi-diameters  of
a  given  ellipse;  tangents  to  the  ellipse  at  P  and  D  meet
at  T:  shew  that  the  triangle  CPT  is  equal  to  the  tri-
angle CDT.
•2.  CP  and  CD  are  any  conjugate  semi-diameters  of  a
given  ellipse ;  K  is  any  point  in  PD,  and  CL  is  the  semi-
diameter  parallel  to  PD :  shew  that  the  triangle  KCL  is  of
constant  area.
'3.  CP  and  CD  are  any  conjugate  semi- diameters  of  a
given  ellipse;  PQ  is  a  chord  drawn  parallel  to  a  fixed
straight  line  :  shew  that  DQ  will  be  parallel  to  a  fixed
straight  line.
•  4.  If  from  the  extremities  of  the  axes  of  an  ellipse  any
four  parallel  straight  lines  be  drawn,  the  points  at  which  they
cut  the  curve  will  be  the  extremities  of  conjugate  diameters.
