Performing  then  the  algebraical  operations  indicated,  there  ob
tains,

—  A  /  =  ^  (-  ^  sin9  <£  +  -  0  sin  </>  cos  $  +  I  —  cos  0  —  <£  sin  <#


--  JJT  (  ^  sin9  tf>  —  0  sin  $  cos  <£  +  cos  0  -f  1  \

~  2  *  sin  *  cos  *  +  sinS  *  +


When  the  arc  </>  is  small,  which  is  usually  the  case  within  the
ordinary  limits  of  practice,  the  preceding  expression,  by  suitable
reductions,  the  details  of  which  cannot  be  entered  into  here,  re-
duces to

"  A  f  =  IM    ^      l  +  a0081


Although  the  preceding  expression  has  been  constructed  under
the  supposition  that  0  is  quite  small,  and  the  arch  very  flat,  in

2  f
which  case  </>  p  =  a,  and  0  =  2  tan  £  <£  =  -~,  it  will  still  give  ap-

proximate  values   of  no   very  considerable   errors  when  0  =  -,
in  which  case/  =  a  =  p,  and  the  preceding  expression  becomes


C"ase  2.  To  find  the  changes  in  the  versed  sine  of  the  curve  when
the  weight,  or  load,  is  uniformly  distributed  over  the  chord,,  or  span
of  the  curve.
In  this  case,  as  the  weight  is  uniformly  distributed  along  the

(*)  For  this  integration  see  Church's  Calcutta,  art.  169,  p.  284,  and  art
190,  p.  264.
618  APPENDIX.
