In  obedience  then  to  the  simultaneous  action  of  these
forces,  the  body  will  receive  the  velocities  A£,  A^,  pro-
portional to  their  intensities,  and  in  the  direction  of  the

forces,  and  these  com-
ponent velocities  will
have  a  resultant  which
will  be  the  diagonal
of  the  parallelogram
Abed.  If  we  take  AB
and  AD  proportional
to  the  velocities  Ab

J[        £ jo >jp    and  Ad  to  represent

Fie.  49.  the  forces  P  and  Q

producing  these  small  velocities,  the  resultant  of  these
forces  to  which  the  resultant  velocity  is  due,  will  be  pro-
portional to  the  velocity  imparted  in  the  same  time  and
in  the  direction  of  its  action,  or  to  Ac ;  we  have,  then,


COMPOSITION   OF   MOTIONS,  VELOCITIES,  AND   FORCES.      135


Then  the  resultant  R  will  be  represented  in  magnitude
and  direction  by  the  diagonal  AC  of  the  parallelogram
ABCD.
Then  the  resultant  of  two  forces  acting  simultaneously
upon  the  same  body  is  represented  in  magnitude  and  di-
rection by  the  diagonal  of  Q
a  parallelogram  construct-
ed upon  these  two  forces.
Reciprocally,  every  force
can  be  resolved  into  two
others,  in  any  two  arbitrary
directions,  equal  to  the
sides  of  the  parallelogram


FIG.  50.
whose  diagonal  is  the  given.
force,  and  whose  sides  are
parallel  to  the  given  directions.
If  the  two  directions  are  perpendicular  to  each  other
we  have


=E  cos  CAB,  Q=R=K  cos  CAD.
