To  determine  this  constant  it  is  only  necessary  to  determine  the  a  for  each
ahape  of  head  fi-om  the  data  furnished  by  practice  at  various  ranges.  This
may  be  done  as  follows : —

Let  om  be  the  angle  of  elevation  found  by  experiment  as  giving  a  range
of  8  feet  to  a  projectile  having  an  ogival  head  of  n  diameters  radius.  We
have

s^L^V.+u^)    and    «„=^  =  ig^,  (4)

whence,  eliminating  On,  we  have

By  plotting  the  function  on  the  right  hand  in  terms  of  x,  the  value  of  x
which  fulfils  the  conditions  of  (5)  can  be  found,  and  substituting  this  value  in
{4)  we  find  a^.  A  comparison  can  then  be  made  between  the  trajectory  found
by  experiment  and  that  given  by  the  formul89(l),  (2),  (3).
I  will  not  add  to  the  length  of  this  paper  by  giving  the  somewhat  tedious
computations  required,  but  the  results  are  shown  in  Diagrams  (II),  (III).
Diagram  II  gives  the  observed  angles  of  elevation  required  at  1000  yards
for  the  various  bullets  tried,  plotted  in  terms  of  the  radius  of  the  ogive
expressed  in  diameters  of  the  projectile.
Diagram  III  shows  the  value  which  must  be  assigned  to  a  if  a  shot  fired  with
velocity  2000,  2200,  etc.,  f.s.  requires  an  angle  of  elevation  a  for  1000  yards.
VOL.  LXXX. — A.  2  0


On  tfie  Trajeciorks  of       [Feb,  24^

Bj  putting  a=^Q  we  find  the  mlnmiuiii  angle  of  elevation  which  a  projectile
at  the  given  range  can  have,  and  the  values  so  foaud  approach  the  angles
actually  tised  with  largo  projectiles  at  ranges  short  enough  to  make  the
reduction  of  velocity  by  air  resietance  inconsiderable.
d  10  iz

Calibnes  radius.
