of  Loaded  and  Unloaded  Bars.  357 
--Integrating  and  determining   M0  and    the  arbitrary  con- 
stants by  the  terminal  conditions 
=  •Q76931^,y1  (.565476  iV_.7U0±8  ^  +  .41g  lv 
shLL 
~-297619^7  +  -059524  a?8), 
and 
_h  =  235-9^. 
A  small  error  in  the  value  of  x  for  which  y—y±  makes  no 
appreciable  difference  in  the  final  result. 
The  next  approximation  gives 
-V  =  '018^I6^1(2,36692  PV  -  3-10194  ZV  + 1-57077  IV 
-  -88577  ZV  +  -08267  ZV° -  -03758  lx11  +  -00493  x12)  ; 
and  putting  a?  =  '5805  I, 
-^  =  238-0  J 
2/i    '  P^ 
and  _       1542  W 
~    ~     2tt      P  ' 
§  5.  Clamped-Free  Bar  of  Circular  Section,  Diameter  vary- 
ing as  distance  from  Free  End. — Taking,  as  previously  in 
similar  cases,  the  origin  at  the  free  end, 
Let 
(9-        ~2\ 
and  the  diameter  d=A.x.     Then 
dz?      EAVjo  *  K       '>\         I   +  PjT" 
-y  =  i|^-l(-016/2--04&+-0416«2— 016y+-0027^Y 
2/i  ^ 
