Constructions. — 1°.  Of  the  piedouche,  and  of  some  preliminary
points  of  shade.  PI.  X.,  Fig.  31.  Assume  the  bases,  H'C'  and
F'G",  of  the  double  curved  portion  of  the  piedouche.  At  F',
take  any  vertical  height  F'E',  greater  than  half  the  distance
between  the  assumed  bases.  Through  E',  draw  the  line  E'l',
parallel  to  the  bases,  and,  from  C',  let  fall  C'D',  perpendicular  to
E'l'.  Then,  with  E'F'  as  a  radius,  describe  the  quadrant  from
F'  to  E'l',  and  with  D'C'  as  a  radius,  describe  the  quadrant  from
ET  to  C'.
The  compound  curve,  thus  formed,  will  be  the  meridian  curve,
whose  revolution  about  a  vertical  axis,  A — T'A',  will  generate
the  double  curved  portion  of  the  piedouche.  Its  cylindrical
bases  are,  as  shown  in  vertical  projection,  above  and  below  the
double  curved  portion.  These  bases  are  horizontally  projected
in  the  concentric  circles,  A — IIYB,  and  A — GWF.
In  applying  the  method  of  (51)  some  of  the  assumed  circles
of  contact  of  the  auxiliary  surface  may  prove  to  be  beyond  the
limits  of  the  curve  of  shade,  so  that  it  is  best  to  construct  the
highest  and  lowest  points  of  that  curve  first,  as  in  (Prob.
XXVII.).
Let  AJ — A'J'  be  a  projection  of  a  ray  of  light.  Eevolve
this  ray  about  the  axis  A — T'A'  to  the  position  AJ" — A'J'"


78  GENERAL    PROBLEMS.
\

Draw  IV  and  EV'"  perpendicular  to  A'J"',  then  u'  and  r'",
horizontally  projected  at  u  and  r",  will  be  revolved  positions  of
the  highest  and  lowest  points.  In  the  counter-revolution,  these
points  return  to  IT,!!'  and  r,r',  which  are  their  real  positions.
