To  complete  the  construction  and  draw  the  metacentre  it  is
necessary  to  determine  the  radius  of  curvature  p  for  each  angle  used
in  locating  the  curve  of  buoyancy  by  aid  of  equation  (19),

/


The  volimie  V  is  of  course  the  constant  of  the  isocarene  and  is  there-
fore known,  but  it  is  necessary  to  locate  the  water-line  WU  correctly
in  order  that  its  moment  of  inertia  /  can  be  determined.  The  most
convenient  way  of  doing  this  is  to  draw  a  third  series  of  cross-curves
for  values  of  T/,  Fig,  70,  at  all  angles  and  displacements ;  this  work
can  be  carried  out  at  the  same  time  that  the  cross-curves  for  OT
and  for  TB'  are  constructed.  Having  the  value  of  p,  it  may  be  laid
off  from  5',  thus  giving  a  point  of  the  metacentric;  after  a  sufficient
number  of  points  are  located  that  curve  can  be  drawn  definitely.
SURFACES  OF  BUOYANCY  AND  OF  PLATER-LINES.
137


Sections  of  the  Surface  of  Buoyancy. — The  path  of  the  centre  of
buoyancy  for  a  very  small  inclination  may  be  assumed  to  lie  in  a
plane  passed  through  the  original
vertical  and  the  new  centreof  buoy-
ancy B\  Fig.  72.  The  curve
BB'  may  be  assumed  to  be  the
arc  of  a  circle  having  its  centre  on
the  original  vertical  BZ.  Let  Fig.
73  represent  the  arc  BB'  revolved
into  the  plane  of  the  paper  and
extended  to  half  a  circle.  The
half -chord   AB'   has    the    length


VaH/J',


Fig.  72.
Fig.  73.
