360  Mr.  J.  Morrow  on  the  Lateral  Vibration 
In  either  case,,  x—a  gives 
3EI/  I 
Pa    _    ;^J1/  ±Y 
ya          m   \ab)  ' 
This  result  and  those  of  §§  7  and  8  are  identical  with  the 
corresponding  results  obtained,  otherwise,  by  Chree. 
§  10.  Supported  Bar  with  two  symmetrically  placed  Concen- 
trated Loads. — When  there  are  two  equal  masses,  placed  at 
equal  distances  a  from  each  end  of  a  bar  whose  mass  is 
negligible,  an  exact  solution  can  be  obtained  in  a  precisely 
similar  manner.     Thus,  for  a  bar  supported  at  each  end, 
For  x  <  a, 
„T  ..      /  x2       la   ,   a2\ 
Ely    =™yax{j  -  9  +2  } 
For  x  =  a  to  (I  — a), 
EIa?=fiiy.«, 
t^t     i  ..      /a2      Ix       x2\ 
EI  y<   =m^a^ --+-). 
The  curve  between  the  two  masses  is  a  circular  arc  of 
radius  El/mj^  a,  and  the  inclination  of  the  extreme  ends  is 
mijal(a-l)IEI. 
V/„  _  6RI 
ya        ma2(4ia  —  'diy 
§11.  Clamped- Clamped  Bar  with  tico  Concentrated  Loads. 
— In  the  previous  case,  if  the  ends  are  clamped,  we  must 
subtract  M0  (i.  e.  the  couple  at  each  end  required  to  fix  the 
d2y 
directions)  from  the  expression  for  ■—, 
We  find 
M0    =^aa{l-a)1 
I 
x2  i  „     ,        Ix 
and 
®iy  =  ™yajl(a2--ia+  3  J, 
ya=       6EH 
?/a      ma\'2l  —  'da)' 
