of  Loaded  and  Unloaded  Bars.  367 
and 
Eh/  =  PoyFi/'a  La>^VlO-6/3%5573^V  +  5-5113^3  +  4-9603/V 
-3-3069/V  +  *5511.r10--3006^  +  -0501^  +  —  «|(3«2Z- 2a3)^ 
+  (an-2a*P)^0+  ^-a^+(52an*-22anS-2a?+7aH)^ 
+  *%-  { (3ft2/  -  ^)  ^  +  (3«3/  ~  6«2/2)  ^  +  3a%  -  a3/3} . 
When  x=a  these  give 
~)     =  ^ET  \p(D%  10-6(3-5573/8-5-5113/7a  +  4-9603Z4a4 
-3-3069ZV-h-5511rt8- -3006  ^+-0501^] 
+  j^p  (3/6-lir'a  +  13ZV-2/3a3-7/V  +  5/a5-aGjl 
a3^3  wi 
+  ~3FeT 
§  20.  As  a  particular  case,  if  the  load  be  at  the  centre 
P-x=  '0020213/rf  +  -0052083mZ3 
and 
~  #T  ~       LI 92  +  2021-3/w»«  +  5208-3m/,<D   J 
If  the  masses  o£  load  and  bar  be  equal  we  have  a  further 
simplification  to 
_,}a_  _  139-65  EI 
ya   ~~        ml3 
Section  V.  Practical  Formulae  for  Loaded  Bars. 
§  21.  The  formulae  obtained  in  the  last  section  for  the 
frequency  of  the  lateral  vibrations  of  loaded  bars  of  appreci- 
able mass  are  too  cumbersome  for  general  use.  In  every 
case,  however,  given  the  position  and  mass  of  the  concentrated 
load  and  the  mass  of  the  bar  (assumed  uniform  throughout 
