Principles. — This  problem,  enunciated  more  as  an  abstract  one,
is  this :  To  construct  the  shadow  of  a  vertical  circle  upon  the
interior  of  a  hemispherical  surface  of  which  the  given  circle  is
the  front  edge.
83.  If  a  straight  line  be  moved,  so  as  to  remain  constantly
parallel  to  its  first  position,  and  to  rest  on  the  circumferences  of
two  great  circles  of  a  sphere,  it  will  generate  a  cylinder,  of  which
those  circles  will  be  equal  oblique  sections.  All  the  elements  of
this  cylinder  will  intersect  the  sphere,  except  the  two  which  are
tangent  to  it  at  the  extremities  of  the  common  diameter  of  the
two  great  circles.  Hence  we  have  the  principle,  that  a  cylinder
which  intersects  a  sphere  in  a  great  circle,  intersects  it  also  in  a  second
great  circle.  The  right  section  of  this  cylinder  will  be  an  ellipse,
equally  inclined  to  these  great  circles,  and  whose  transverse  axis
is  their  common  diameter.
It  now  follows,  that  the  cylinder  of  rays  passing  through  the
front  edge  of  the  quarter  sphere,  will  intersect  its  interior  in  an
arc  of  a  great  circle  of  that  sphere,  which  arc  will  be  the  shadow
required.
1°.  Tlie  first  solution,  based  on  the  foregoing  general  princi-
ples, is  made  by  the  direct  general  method  (82),  and  depends  imme-
diately on  the  following —

Principles. — A  plane  of  rays,  perpendicular  to  the  vertical  plane
of  projection,  will  cut  a  circular  arc  from  the  spherical  part  of  the
niche,  and  a  ray  from  the  cylinder  of  rays,  whose  intersection  will
be  a  point  of  the  required  shadow.
Construction. — PI.  X.,  Fig.  33.  In  this  construction,  three
planes  of  projection  are  used.
