*  Lagrange's  Method,  Routh,  Rigid  Dynamics,  Vol.  i.,  p.  369.
106]  HARMONIC    SCREWS.  99

106.     Discussion  of  the  Results.
For  the  position  of  the  body  before  its  displacement  to  have  been  one  of
stable  equilibrium,  it  is  manifest  that  the  co-ordinates  must  not  increase
indefinitely  with  the  time,  and  therefore  all  the  values  of  s2  must  be  essen
tially  positive,  since  otherwise  the  values  of  #/,  ..  0n'  would  contain  expo
nential  terms.
The  2n  arbitrary  constants  are  to  be  determined  by  the  initial  circum
stances.  The  initial  displacement  is  to  be  resolved  into  n  twists  about  the  n
screws  of  reference  (§  95).  This  will  provide  n  equations,  by  making  t  =  0,
and  substituting  for  Oi,...0n',  in  the  equations  just  mentioned,  the  amplitudes
of  the  initial  twists.  The  initial  twisting  motion  is  also  to  be  resolved  into
twisting  motions  about  the  n  screws  of  reference.  The  twist  velocities  of

JQ I  JQ    >

these  components  will  be  the  values  of  — ' ;  ,   . . . ,  ~,™  ,  when  t  =  0  ;  whence

we  have  n  more  equations  to  complete  the  determination  of  the  arbitrary
constants.
If  the  initial  circumstances  be  such  that  the  constants  H.,,  ...,  Hn  are  all
zero,  then  the  equations  assume  a  simple  form :

0i  =fu  #!  sin  (s^  +  c),


On'  =/!„£?!  sin  (sj  +  c).
