Motion  by  Impa£l  and  Gravity.  367 
have  pv*=pv*+  '—~r~ . For  fince  v is  the  velocity  of  q_ 
at  a certain  period  of  its  defcent,  and  is  to  the  velocity 
of  any  prifmatic  particle  in  the  body  as  the  diftance  ni d 
from  the  axis  to  the  diftance  of  that  particle  from  the 
fame,  it  is  evident  that  -p—  will  truly  reprefent  the  fum 
of  all  the  particles  multiplied  by  the  fquares  of  their  ve- 
locities. v1  is  therefore  to  vz  as  ap+b  p to  ap , and  the 
whole  force  of  gravity  is  to  the  force  which  accelerates 
the  motion  of  Qjn  the  fame  ratio,  becaufe  in  uniformly 
accelerated  motions,  when  the  Ipaces  defcribed  are  the 
fame,  the  accelerating  forces  are  in  the  duplicate  ratio  of 
the  velocities.  It  is  obvious,  that  the  motion  of  Q_is  uni- 
formly accelerated,  becaufe  the  velocity  acquired  by  any 
defcent  is  to  the  velocity  of  any  point  in  the  body  always 
in  the  fame  ratio ; and  therefore  the  adtion  of  coupon  the 
body  is  the  fame  as  if  both  were  at  reft.  Farther,  the 
altitude  z through  which  a heavy  body  muft  fall  to  ac- 
Cl  1)L 
quire  the  velocity  v is  plainly  equal  to  R x — - --,  for.  the 
altitudes  z and  r are  inverfely,  as  the  forces  which  gene- 
rate the  equal  velocities.  Laftly,  the  time  of  qJs  defcent. 
is  equal  to  t x ; beeaufe  the  times  are  always  in 
the  fub-duplicate  ratio  of  the  fpaces  diredtly,  and  forces 
inverfely. 
Ifc 
