We  see,  therefore,  that  for  a  ship  to  be  in  stable  equilibrium
for  any  direction  of  inclination,  it  is  necessary  that  the  meta-
centre  be  above  the  centre  of  gravity  of  the  ship.  We  now
group  together  the  three  conditions  which  must  be  fulfilled  in
order  that  a  ship  may  float  freely  and  at  rest  in  stable  equili-
brium—

(i)  The  weight  of  water  displaced  must  equal  the  total
weight  of  the  ship  (see  p.  21).
Conditions  of  Equilibrium,  Transverse  Metacentre,  etc.  93

(2)  The  centre  of  gravity  of  the  ship  must  be  in  the  same
vertical  line  as  the  centre  of  gravity  of  the  displaced  water
(centre  of  buoyancy)  (see  p.  89).
(3)  The  centre  of  gravity  of  the  ship  must  be  below  the
metacentre.
For  small  transverse  inclinations,  M  is  termed  the  transverse
metacentre,  which  we  may  accordingly  define  as  follows  : —

For  a  given  plane  of  flotation  of  a  vessel  in  the  upright
condition,  let  B  be  the  centre  of  buoyancy,  and  BM  the  vertical
through  it.  Suppose  the  vessel  inclined  transversely  through
a  very  small  angle,  retaining  the  same  volume  of  displacement,
B'  being  the  new  centre  of  buoyancy,  and  B'M  the  vertical
through  it,  meeting  BM  in  M.  Then  this  point  of  intersection,
M,  is  termed  the  transverse  metacentre.
There  are  two  things  in  this  definition  that  should  be  noted :
(i)  the  angle  of  inclination  is  supposed  very  small,  and  (2)  the
volume  of  displacement  remains  the  same.
It  is  found  that,  for  all  practical  purposes,  in  ordinary  ships
the  point  M  does  not  change  in  position  for  inclinations  up  to
as  large  as  10°  to  15°;  but  beyond  this  it  takes  up  different
positions.
