We  have  first  to  prove  that  the  five  elements  which  can  be  thence
inferred  with  respect  to  the  rigid  body  are  in  general —

(1)  A  diameter  of  the  momental  ellipsoid.
This  is  clearly  equivalent  to  two  elements,  inasmuch  as  it  restricts
the  position  of  the  centre  of  gravity  to  a  determinate  straight  line.
(2)  The  radius  of  gyration  about  this  diameter.
This  is,  of  course,  one  element.
(3)  A  straight  line  in  the  plane  conjugate  to  that  diameter.
A  point  in  the  plane  would  have  been  one  element,  but  a  straight
line  in  the  plane  is  equivalent  to  two.  If  the  centre  of  gravity  were
also  known,  we  should  at  once  be  able  to  draw  the  conjugate  plane.
Draw  a  plane  through  both  the  instantaneous  screw  a  and  the  common
perpendicular  to  a  and  77.  Then  the  centre  of  gravity  of  the  rigid  body
must  lie  in  that  plane  (§  301).  It  was  also  shown  that  if  pa  be  the  pitch  of
a,  and  if  (ar))  represent  the  angle  between  a  and  77,  then  the  perpendicular
distance  of  the  centre  of  gravity  from  a.  will  be  expressed  by  pa  tan  ((277)
(§  301).  This  expression  is  completely  known  since  a  and  77  are  known.
Thus  we  find  that  the  centre  of  gravity  must  lie  in  a  determinate  ray
parallel  to  a.  There  will  be  no  ambiguity  as  to  the  side  on  which  this
straight  line  lies  if  it  be  observed  that  it  must  pass  between  a  and  the  point
in  which  77  is  met  by  the  common  perpendicular  to  77  and  a.  In  this  manner
from  knowing  a  and  77  we  discover  a  diameter  of  the  momental  ellipsoid.
