of  Loaded  and  Unloaded  Bavx .  359 
where  I  is  the  distance  between  the  load  and  the  fixed  end. 
_  //,  =  3ET 
y1     '   mi:6' 
On  the  other  side  o£  the  load  the  bar  is  straight. 
§  8.  Bar   supported  at   Both    Ends. — Origin   at  one  end. 
Mass  divides  length  of  bar  into  segments  a  and  b. 
Here,  for  x<a, 
j^Td2u        o     .. 
Eljx?  =  i  "v  ■•■'■> 
whilst,  for  x  >a, 
The  constants  of  integration,  expressing  the  inclinations  of 
the  bar  at  its  ends,  are  obtained  from  the  consideration  that 
both  -,---  and  y  must  have  the  same  value  in  each  equation 
when  x~a.     The  inclinations  are,  therefore, 
mya-ab(a    i    by 
~  ITT 
Hence 
my.,  b   .  o      ~,    ,         0  N 
y  =  6BTI  /   ^  ~  +       " 
and 
.      mi/a  l  —  Xf*     0    7     ,       o. 
y    6M  ~r~  ^  -"2^+«r)' 
In  either  case, 
_#     _  3EI/ 
y         md2b'2' 
§9.   Clamped- Clamped  Bar. — With  the  same  notation  ai= 
in  the  preceding  case. 
If   cT<«, 
and  if  x>a  we  must  add  mya  (x—d)  to  the  above. 
By  integration  we  find 
tv,t  •  •  «&2  !     t.--  ..   a2b 
M0  =  i»ya-p*     and     Mx  =  mya  ^  , 
-m^^^^ail-aO-bx] 
(a       b\  ,  mw"   a /a2      Z2\ 
and 
^EIy'  =  ~|«~  [(a  +  3&)a?3+3(<i-2&)fo2  +  3^-a/3]* 
