lei  to  A  Y,  and  let  Q  denote  the  intersection  of  the  diameter
A'X'  and  the  ordinate  FM.    Then,  from  the  figure,  we  have

x  =  AM=  AH-i-  A'M'  +  M'Q

=  h^x'  -{-  PM'  cos  PM'Q
=  h  -\-  x'  -{-  y'  cos  dy  (a)

and  y  =  PM=A'H+PQ

=  ]c-\-  PM'  sin  PM'Q

=  k  ^  y'  sin  0.  {b)

But  since  the  point  {x,  y)  is  on  the  curve,

y'  =  2px.  (c)

Substituting  the  values  of  x  and  y  as  given  by  (a)  and  (b)
in  (c),  we  have

(k  4-  2/'  sin  ey  =  2p(h  +  x'  +  y'  cos  ^);
whence

y"  sin'  0  +  22/'(^  sin  6^  -i?  cos  6)-^  ¥  -  2jjh  =  22)x\

But,  by  (6)  of  §101,

Jc  =p  cot  6;

and  since  the  point  {h,  Jc)  is  on  the  curve

F  =  2ph,
therefore  we  have

y'^  sin'  0  =  2px' ,

which  is  the  equation  of  the  curve  referred  to  the  new  axes.
Cor.     This  equation  may  also  be  expressed  in  terms  of
A'F,  the  focal  distance  of  the  point  A\    Thus,  by  (3)  of  §99,

A'F=ip-\-h,

k*
and  k^  =  2ph,        or        h  =  ^.
