was  actually  the  case.  In  the  same  way  we  conclude  that  certain
planets,  travelling  in  their  orbits,  rotate  round  their  axes  though
the  fact  has  not  yet  been  rerified.  Some  equations  must  therefore
tend  to  destroy  all  progressive  motion,  and  others  all  motion  o±  ro-
tation.    How  many  equations  of  each  kind  must  there  »e  .'

It  is  clear  that,  to  keep  a  body  motionless,  it  must  be  hindered
from  moving  according  to  three  axes  in  different  planes— commonly
supposed  to  be  perpendicular  to  each  other.  If  a  body  can  not
move  from  north  to  south,  nor  the  reverse  ;  nor  from  east  to  west,
nor  the  reverse ;  nor  up,  nor  down,  it  is  clear  that  it  can  not  move
at  all.  Movement  in  any  intermediate  direction  might  be  conceived
of  as  partial  progression  in  one  of  these,  and  is  therefore  impossi-
ble. On  the  other  hand,  we  can  not  reckon  fewer  than  three  in-
dependent elementary  motions ;  for  the  body  might  move  in  the
direction  of  one  of  the  axes,  without  having  any  translation  in  the
direction  of  either  of  the  others.  Thus  we  see  that,  in  a  general
way,  three  equations  are  necessary,  and  three  are  sufficient  to  estab-
lish the  absence  of  translation ;  each  being  specially  adapted  to
destroy  one  of  the  three  progressive  motions  of  which  the  body  is
capable.  The  same  view  presents  itself  with  regard  to  the  other
motion — of  rotation.  The  mechanical  conception  is  more  compli-
cated ;  but  it  is  true,  as  in  the  simpler  case,  that  motion  is  possible
in  only  three  directions — in  three  co-ordinated  planes,  or  round
three  axes.  Three  equations  are  necessary  and  sufficient  here  also ;
and  thus  we  have  six  which  are  indispensable  and  sufficient  to  stop
all  motion  whatever.
