Velocities  of  Cannon  Balls , See.  61 
Again,  fince  f is  the  verfed  fine  of  the  deferibed  arc  c , 
. . , r cc  cc  bkk-\-  pht> 
its  radius  being  r;  therefore  as  r : H ::  — : — - x = 
' Ol 
the  verfed  fine  to  the  radius  h,  or  the  verfed  fine  of  the  arc 
deferibed  by  the  center  of  ofcillation,  which  call  v ; then  is 
v the  perpendicular  height  defeended  by  this  center,  and 
the  velocity  it  acquires  by  the  defeent  through  this  fpace 
is  thus  eafily  found,  viz.  as  s/ : V v ::  32^ 
8.02  c 
— 7—  X 
rV  2 
V 
bkk  -f  ghp 
=the  velocity  of  the  center  of  ofcillation  deduced 
from  the  chord  of  the  arc  which  is  actually  deferibed. 
Having  thus  obtained  two  different  exprelfions  for  the 
velocity  of  this  center,  independent  of  each  other,  let  an 
equation  be  made  of  them,  and  it  will  exprefs  the  rela- 
tion of  the  feveral  quantities  in  the  queltion ; thus  then 
8.02  c Ibkk  -\-ghp  f . 
=■-— v / "L ", — 7 ' trom  which  we  obtain  v = 
rv  2 \ " 
we  have  7 
bkv 
ok  -f  gp 
bk+gp 
8.02  C 
bkrV'l 
'Jbk  + gp  x bkk  + ghp  the  true  expreffion  for  the  ori- 
ginal velocity  of  the  ball  the  moment  before  it  ftruck 
the  pendulum. 
corollary.  But  this  theorem  may  be  reduced  to  a 
form  much  more  fimple  and  fit  for  ufe,  and  yet  be  fuffi- 
ciently  near  the  truth.  Thus,  let  the  root  of  the  com- 
pound factor  V bk+gp  x bkk+gbp  be  extracted,  and  it  will 
be  equal  to  Vhxpg+bkx  -gp  within  the  roooooth  part 
of 
