This  volume  is  also  V0  (1  +  kt~).     But  V0  =  11.     Hence

1  +  kt  =  (1  +  at)3  =  1  +  Zat  +  BaH*  +  aH\


EXPANSION.  27

But  since  a  is  a  very  small  quantity,  its  higher  powers
may  be  neglected  in  comparison  with  the  first,  or

1  +  Jet  =  1  +  Sat,
and  Jc  =  Sa  nearly.
This  relation  assumes  that  the  body  is  isotropic,  or  has
the  same  physical  properties  and  expands  equally  in  all
directions.  In  the  case  of  crystals  this  is  true  only  for
those  of  the  regular  cubic  system,  which  do  not  cause
double  refraction  of  light  (I.,  226).  These  dilate  uniformly
in  all  directions  in  the  same  manner  as  amorphous  bodies.
In  general  crystals  have  three  rectangular  axes  of  dilata-
tion, and  the  linear  coefficients  in  these  three  directions
are  not  identical ;  the  voluminal  coefficient  is  then  equal  to
the  sum  of  the  three  linear  coefficients.  It  follows  that  a
crystalline  sphere  at  one  temperature  ceases  to  be  spherical
at  any  other  temperature,  and  a  cubical  portion  of  a  crys-
talline body  at  one  temperature  will  not  remain  cubical
when  the  temperature  changes,  unless  the  crystal  belongs
to  the  cubic  system.
Crystals  belonging  to  the  rhombic  system  have  an  axis
of  crystalline  symmetry,  and  the  two  coefficients  of  expan-
sion perpendicular  to  this  axis  are  equal,  or  the  crystal  has
the  same  properties  in  all  directions  perpendicular  to"  the
axis  of  symmetry.     In  this  case

k  =  ax  +  2a2  •

Here  ax  is  the  coefficient  of  expansion  parallel  to  the  axis,
and  a2  is  the  coefficient  perpendicular  to  it.
